Can you use martial weapons while untrained?

Question

Can you use martial weapons while untrained?

For example, alchemical bombs are martial ranged weapons. Can an untrained character buy and throw these? If so, what modifiers apply to the attack roll?

Answer 1

Can an untrained character buy [these]

Proficiency does not influence whether or not you can buy an item. You may freely buy items no matter what degree of proficiency you have with them. However, you must have access to the item you want to buy. This is explain on page 13 of the Core Rulebook in the "rarity" entry:

Uncommon items are available only to those who have special training, grew up in a certain culture, or come from a particular part of the world. Rare items are almost impossible to find and are usually given out only by the GM, while unique ones are literally one-of-a-kind in a game.

The rarity of an item is described in its entry in the CRB. Items with no rarity information are presumed to be common. Alchemical bombs in general seem to be common, but look them up the specific kind of bomb you want in the Crafting and Treasure section of the CRB just to be sure.

Can an untrained character ... throw these?

Yes. Your proficiency does not determine whether or not you can use an item. You may freely use an item regardless what degree of proficiency you have, including being untrained.

If so, what modifiers apply to the attack roll?

The formula for a ranged attack roll is on page 278 of the Core Rulebook.

Ranged attack modifier = Dexterity modifier + proficiency bonus + other bonuses + penalties.

Your proficiency bonus while untrained is +0. Therefore, your attack roll for using a bomb if you are untrained is 1d20 + Dexterity modifier + 0 + other bonuses or penalties. Needless to say, proficiency is practically a big deal when attempting to Strike, unless you happen to be facing down a hoard of low challenge creatures.