Retrieving a vial or flask and attacking with it requires two object interactions
Attack limits
To draw two flasks and attack with both of them you would need to be able to take four object interactions, two to draw them both, and two to throw them both. You however only have three object interactions
- your free object interaction
- your Fast Hands Use an Object bonus action
- your Use an Object action
so you will not be able to draw two objects and attack with both.
Detail explanations and citations
Throwing a vial of acid, oil or alchemist fire requires an attack with an improvised weapon. In the case of these items, even though you are making an attack, this uses the Use an Object action (see details in the linked Q&A):
Acid. As an action, you can splash the contents of this vial (...)
Alchemist's Fire. (...) As an action, you can throw this flask up to 20 feet (...)
Oil. (...) As an action, you can splash the oil in this flask onto a creature (...)
The text does not state that you retrieve and throw the flask or vial, so you are spending your action exlusively to attack with it. As it is a Use an Object action, you can also use your bonus action for it, due to Fast Hands.
Retrieving an object like a flask or a vial requires its own object interaction. The Use an Object action on page 193 PHB states:
You normally interact with an object while doing something else, such as when you draw a sword as part of an attack. When an object requires your action for its use, you take the Use an Object action. This action is also useful when you want to interact with more than
one object on your turn.
So you are making an attack with the object, why can't you draw it like your sword, as part of the attack? You can, but the rule for getting this free object interaction to draw the weapon with the attack is limited to one a turn. This is explained in more detail on page 190 PHB:
You can also interact with one object or feature of the environment for free, during either your move or your action. For example, you could open a door during your move as you stride toward a foe, or you could draw your weapon as part of the same action you use to attack.
If you want to interact with a second object, you need to use your action. (...) The DM might require you to use an action for any of these activities when it needs special care or when it presents an unusual obstacle.
(I think this all would be simpler to explain if the PHB just would have said that interaction with a non-magical object not covered under another actions means you take the Use an Object action, and that you can use one such action for free each turn.)