Well, let's look at the feat:
You can cast many spells without needing to utilize minor material components.
Benefit: You can cast any spell with a material component costing 1 gp or less without needing that component. The casting of the spell still provokes attacks of opportunity as normal. If the spell requires a material component that costs more than 1 gp, you must have the material component on hand to cast the spell, as normal.
There are no prerequisites, so yes, he could take the feat. However, let's look at the Alchemist's actual class features. The Alchemy class feature specifically says:
Although the alchemist doesn't actually cast spells he does have a formulae list that determines what extracts he can create.
So while he can take the feat, the feat wouldn't actually do anything for him, since extracts are not spells.
Even if you can take the feat you would still need an alchemist kit to create your bombs, extracts, and mutagens. At best all it would do would be to remove the need for low cost material components, but you would still need the base chemicals to have something to drink. So you might be able to make an extract of Spider Climb without having to have a live spider, but would still need the base chemical and container.