Alchemist Items have a rarity just like any other item
It might not be listed in the Adventurer's Vault but all items in 4e were overhauled to have an Item Rarity of Common, Uncommon, and Rare. Common Items can be easily found and purchased. The item rarity rules can be found in the free, downloadable Errata. (The Adventurer's Vault rarity updates are found in the last couple of pages, for instance.) You can look up an Item's rarity quite easily using the D&D 4e Online Compendium. To read the item description you would need an Insider account subscription, but you can find out the rarity without it: you can search freely, and Rarity is one of the columns.
Yes, anyone can purchase Alchemist's fire.
A quick search shows that Alchemist's Fire is common and thus they should be able to find and buy it. Use your judgement as a GM, a small rural village probably won't have it, but a bustling city should have it in stock somewhere.
