No. They might be able to try, whereas someone wihtout proficiency in either wouldn't even be able to make the attempt. But this will most likely result in a dead Herbalist and a perfectly safe intended victim.
Let's look at the rulebooks.
TOOLS (Player's Handbook, pg 154)
A tool helps you to do something you couldn't otherwise do, such as craft or repair an item, forge a document, or pick a lock. ... Proficiency with a tool allows you to add your proficiency bonus to any ability check you make using that tool. Tool use is not tied to a single ability, since proficiency with a tool represents broader knowledge of its use.
Proficiency with a tool/kit means you have gained the requisite knowledge, experience, and training to do specific things with them. As to why can't this apply to whatever you want, the DMG elaborates:
TOOLS (Dungeon Master's Guide pg 239)
Having proficiency with a tool allows you to apply your proficiency bonus to an ability check you make using that tool. For example, a character proficient with carpenter's tools can apply his or her proficiency bonus to a Dexterity check to craft a wooden flute, an Intelligence check to craft a wooden secret door, or a Strength check to build a working trebuchet. However, the proficiency bonus wouldn't apply to an ability check made to identify unsafe wooden construction or to discern the origin of a crafted item, since neither check requires tool use.
So we can infer from this that proficiency with Carpenter's Tools does necessarily impart knowledge of History (origins of an item) or Investigation/Perception (is this bridge sturdy?). It lets the character do things they've been trained to do: craft wooden objects. And the components of a set of artisan's tools are specialized: Carpenter's Tools and Smith's Tools each include a "hammer", but a hammer best suited for tapping a woodworking chisel is not going to work well if you try to use it to forge-weld metal.
In the same way, an Herbalism Kit and Poisoner's Kit have some similar components, but not identical, and not as specialized. And while not all tool proficiencies don't imply knowledge of history or arcana, some explicitly do as part of the training required to become proficient. Each explicitly says what you can use them for, and what they can help a properly trained character to attempt to do. Xanathar's Guide to Everything gives more specific detail into these two, and let's also include Alchemist's Supplies, as they're very similar as well. Here are the (heavily truncated) descriptions:
ALCHEMIST'S SUPPLIES (XGE pg 79)
...enable a character to produce useful concoctions, such as acid or alchemist’s fire.
- Components. glass beakers, metal frame (hold beaker over flame), glass stirring rod, mortar & pestle, pouch of alchemical ingredients (salt, powdered iron, purified water)
- Arcana. more information on Arcana checks involving potions and similar materials
- Investigation. add insight into any chemicals or other substances...used in an area
- Alchemical Crafting. create alchemical items: dose of acid, alchemist’s fire, antitoxin, oil, perfume, soap
- Activities. Create a puff of thick smoke (DC10); Identify a poison (DC10); Identify a substance (DC15); Start a fire (DC15); Neutralize acid (DC20)
HERBALISM KIT (XGE pg 82)
... allows you to identify plants and safely collect their useful elements.
- Components. pouches, clippers & gloves for collect plants, mortar & pestle, glass jars
- Arcana. add insight to magical studies that deal with plants, to identify potions
- Investigation. details and clues others might miss [in] an area overgrown with plants
- Medicine. improve ability to treat illnesses and wounds...with medicinal plants
- Nature/Survival. in the wild, identify plants and sources of food
- Identify Plants. with a quick inspection of their appearance and smell
- Activities. Find plants (DC15); Identify poison (DC20)
POISONER'S KIT (XGE pg 83)
... allows you to apply poisons and create them from various materials; helps you treat them.
- Components. glass vials, mortar & pestle, chemicals, glass stirring rod
- History. recall facts about infamous poisonings
- Investigation/Perception. inspect poisoned objects or try to extract clues from events that involve poison
- Medicine. treat the victim of a poison; insight into how to provide the best care
- Nature/Survival. acquire lore about which plants and animals are poisonous
- Handle Poison. handle & apply poison without risk of exposing yourself to its effects
- Activities. Spot a poisoned object (DC10); Determine the effects of a poison (DC20)
Each Kit discusses working with very specific things for very specific purposes:
- Alchemy: "chemicals" and "substances", states you can create specific potions (of utility), involving same chemicals and substances (salt, iron, purified water, chemicals, acid)
- Herbalism: "(medicinal) plants" and "herbs"; potions, but specifies "healing potions"; treatment of illness/wounds, but specifies "with medicinal plants"
- Poisoner: "poisons", also "substances", "chemicals", and "materials" (pertaining to poison); "plants" and "animals" that are poisonous; and treament, application, and creation of poisons.
The first two can "identify" a poison, but one proficient with a Poisoner's Kit can "determine the effects of a poison", "recall facts about infamous poisons", "inspect poisoned objects", and "extract clues". Most importantly, only the Poisoner's Kit implies how to SAFELY handle, apply, create, collect, or treat poisons specifically. A doctor can diagnose luekemia, but an oncologist knows much more about how to prevent, treat, and manage it.
I'd say that proficiency in one of these three could possibly allow a character to attempt to do what the others can, but certainly not with ease, proficiency, experience, or any semblance of safety, to them or their tools. I dunno about you, but I wouldn't want someone healing my to be grinding up herbs in a mortar & pestle that was used last night to mix snake venom and neurotoxin; or to quaff a healing potion poured into the same vial that previously held acid. As always, though, ask your DM. They'll decide if you can even try it, and how difficult or dangerous will be the attempt, if any.
Elaborate, Long-Winded Answer
(spurred on by pedantry, insomnia, & too much caffeinne)
Proficiency with Alchemist's Supplies means knowing about and how to use materials and substances, like salt, iron, purified water, etc... In modern terms, a Chemist. They make "useful" potions (utility), not healing or magical. They deal with basic, fundamental things, like acids, fire, smoke, stone, charcoal, etc. They can identify a poison or potion by investigating their ingredients, they are not proficient with either.
Proficiency with an Herbalism Kit means knowing about medicinal plants and herbs--those specific to the purpose of healing. In modern terms, a Botanist or Healer. You can make potions and salves that heal or treat injury or illness. You might identify a poison, even know what a certain poison does to a body, but you don't know how or where to collect, craft, or administer them. You can't create acids or cause alchemical reactions.
Proficiency with a Poisoner's Kit means you can also deals with plants a bit, but the expertise is with poisonous ones. You also know about poisonous animals and materials, how it might have been administered, how much was needed, etc.You have the tools to collect and increase toxicity of these substances, and know how best to apply and administer poisons, as well as extensive knowledge and training in their history and use.
An Alchemist might know that a poison involves chemicals X, Y, and Z; but not how or where to best get it or how it was adminstered. They can isolate and combine ingredients to make "useful concoctions", but that's alchemy (chemistry). An Herbalist might know that a poison causes this pattern of necrosis or rash, but not where it comes from, how to isolate it, or whether it was natural (stepped on a thorn) or administered. They can find and combine plants to reduce its effect, but that's healing (botany).
Ah, but the Poisoner, the Poisoner can tell how the victim was poisoned, if it's natural or crafted, what the ingredients are, where to find them, their rarity or cost, substitutions. Whether the poison is a unique coctail used by a specific Guild thought destroyed a century ago. They have the tools and training to, say, find, catch, and milk a snake's venom glands without being bit themselves. And they will be better at treating poisoned victims than the Herbalist or Alchemist...if they want to. They might also make a treatable poisoning lethal, and the Herbalist or Alchemist won't be able to tell.
Proficiency in a set of tools does not imply proficiency in every related Skills, nor vice-versa:
- a Fighter is lethal with a sword, but has to seek out a smith to craft or repair one
- a Ranger knows exactly where an herb grows, have not which part they need, or how to use it
- an Alchemist might know a type of herb contains a chemical, but not what it looks like, that it must be kept cold, or harvested in spring
- an Assassin might know certain salts are used black powder, but not stabilizers (like starches) to keep a pistol from exploding
- a Thief might know the names of poisons, and which killed a lord in their city, but not how to actually make and manage it (or use it without getting caught)
To answer your question, no, Herbalism Kit proficiency does not grant imply knowledge of poisonous plants, animals, or substances, only medicinal plants and herbs. Our Healer can identify a plant with a cursory examination, point out a poisonous vine about to be stepped on, and what symptoms a poison causes, but not how to safely or reliably find, extract, isolate, administer, or even treat said poison. It's not just the tools that matter, it's the specific knowledge, history, practice, and expertise that comes with being trained in their use. Without Nature proficiency, they've no clue where to begin to look for it in the wild.
I'd think any of these three proficiencies (Alchemy, Herbalism, Poisoner) would allow a character to attempt (ATTEMPT) to do what the others can, but not with the bonuses the correct training, experience, and knowledge would impart, nor would they know where to get the right ingredients, or how. And not at all safely, effectively, or without contaminating or destroying their equipment. Possibly adding INT or WIS mod, but no proficiency, at disadvantage and/or Difficulty Check. This would, of course, depend on your DM, and if you can make a convincing case for at least a passing familiarity with the process: "My brother, at the monestary? He was a ninja... (also my dad's a king and I know karate)". (roll Deception)
So, could our Herbalist create a poison? Maybe. Arguably, better than somebody who can't tell a potted plant from a painting. But note that a Healer's Kit doesn't talk about knowing anything about poisonous plants, how to safely handle all plants, how to combine, or mix, or administer them. The components don't include multiple glass beakers or stirring rods you'd need to isolate, concentrate, or amplify the toxicity of the plants they know, just how powder and combine them. You want to create a few drops of odorless venom? You create what looks like a chewed-up wad of moss the size of your fist that stinks of bogwater.
You're more likely to give someone a bellyache then a heart attack. Maybe you do manage to get someone sick, maybe even severely, but it won't be as potent (or discreet) as what a Poisoner could do. And our Herbalist is just as likely to end up poisoning themselves in the process, damaging/poisoning their equipment, or clumsily slopping a salve into a drink that is only poisonous if injected, and anyone with any awareness will see a floating blob of muck and probably retaliate.
Bottom line: If you wanna make poisons, get trained and proficient with a Poisoner's Kit.