I was wondering if it would be possible for a level 1 character to start with a magic item if it wasn't a campaign requirement. I am the DM. Is it possible?
Nothing in the rules prevents it, but they suggest to give none
The DMG does not contain any explicit rule or indication preventing this possibility. Indeed, the Magic Item section of the DMG, page 135, contains the following table:
|Common||1st or higher||50-100 gp|
|Uncommon||1st or higher||101-500 gp|
|Rare||5th or higher||501-5000 gp|
|Very Rare||11th or higher||5001-50000 gp|
|Legendary||17th or higher||50000+ gp|
Hence, the official rules allow common and even uncommon items at level 1.
The above table follows the description of Tier 1 in the DMG.
At the start of their careers, characters use 1st- a nd 2nd-level spells and wield mundane gear. The magic items they find include common consumable items (potions and scrolls) and a very few uncommon permanent items.
Pay attention that the above description says "The magic items they find [...]": it seems to suggest that magic items are found during adventures.
Moreover, there is a table1 in the DMG, page 38, that serves as a guideline for starting equipment for campaigns at higher levels, for each tier: in particular, for tier 1 it suggests normal starting equipment, even in high magic settings. In my opinion, starting at 4th level without any magic item seems quite contradictory: indeed, the same DMG tells us (emphasis mine):
Starting equipment for characters above 1st level is entirely at your discretion, since you give out treasure at your own pace. That said, you can use the Starting Equipment table as a guide.
In conclusion, I see the table at page 38 more as a guideline rather than a rule, leaving to the DM the decision (as 99% of edge cases of this type).
Since you are the DM, you have the complete freedom by Rule 0: in case you decide to provide first level characters with magic items, think really carefully which ones you are giving.
The first level seems to be randomly deadly (see for example this answer) and designing balanced encounters using the CR system doesn't always help (see this Q&A). Providing the characters with a common (or uncommon) magic item from the beginning of the campaign could be not game breaking, but as a DM you should be really careful in which items you let them pick.
For example, a potion of healing is not unbalancing at all, it could be a life-saver in case the players roll poorly and the enemies are very lucky. An adamantine armor could be another life-saver, but beware to not give a full plate armor. Other consumables, such as Ammunition +1, could be a good choice, maybe not in great amounts.
On the other hand, there are some uncommon items that could be game breaking, if not used (by the DM) with particular attention. Consider the Elemental Gem: it allows to cast the Conjure Elemental spell, which could conjure an elemental up to CR 6. If used improperly, it could turn the final encounter with the boss of the party's first adventure into Netflix'n'chill.
1 Credits to TreeSpawned that in their answer pointed out this table, which I completely missed in the first version.
The rules about starting equipment don't endorse it
Of course, you as the DM can decide what ever you want for your players in your new campaign. However if you are asking what the official rules about starting equipment have to say on that matter, take a look at the table on page 38 of the DMG. Here you can find suggestions for starting equipment for characters by level for low, standard and high magic campaigns.
For high magic campaigns, the table suggest adding one uncommon magic item to the starting equipment of characters starting at levels 5-8, but none for characters starting at level 1-4. One could argue this table is only for "Starting at higher level", as the headline suggest, however they did make a conscious effort to include the first tier of play and wrote the default starting equipment in all three columns, clearly implying that the default starting equipment for a low level character shouldn't include any magic items. Keep in mind that this table is for "experienced players", as the paragraph above explaines, and even they should apparently not get any magic items at level 1.
Of course as with all "rules" in D&D, these are just guide lines. If you think it would make the game more fun for you and your group to add a free magic item at level 1, feel free to give it a try. I personally would suggest giving them a common magic item, as these normally don't have much of an impact on a character's strength and are usually just some interesting tools for role playing. You can find a whole bunch of them in Xanathar's Guide to Everything, if you are interested.
Yes, with the correct Class, Race, and/or Background abilities.
There are classes like the Forge Cleric, Artificer, and Pact of the Blade Warlock that have the ability to imbue non-magical weapons and armor with a magical +1 bonus to hit and damage (in the case of the Warlock, this can only affect their Pact Weapon). Additionally, there are some races like the Mark of Making Human that can innately cast the magic weapon spell. Of these, only the Forge Cleric and Mark of Making Human get the magic weapon from level 1.
I'm not listing the Hexblade's Hex Warrior ability, because while they do have the ability to "mystically channel your will through a particular weapon", this doesn't actually make it count as a magic weapon.
Additionally, while they aren't magic weapons, there are several Trinkets that are explicitly magical, like the cupcake that magically restores itself everyday from the Feywild Trinkets list. Similarly, the Genie Warlock gets a magical vessel as part of their subclass abilities.
Finally, in the Adventurer's League - at least at one point, it's possible that its been changed since the last time I checked - if you took the Inheritor background, the benefit of your Inheritance was defined as being a single item that costs no more than 50gp that could have a minor magical effect attached to it like glowing with the brightness of a candle. In a game that isn't limited that way, it could potentially be anything, and the background actually mentions magic items as a possibility.