There is no difference between conjugation and possessive in this regard. The formal/polite way is the 3rd person in all cases. Although I'm not sure I'd say "Van táskája?", rather something like "Táska kapható Önöknél?" (kapható = available/buyable/for sale)
The question is whether you want to use the informal or the formal way of addressing the person, what register you want to use, etc. (3rd person = formal, 2nd person= informal)
That also depends on your age and the shop assistant's age, whether you know him/her, whether you want to give the impression of being somebody relatively well educated/polite or not, etc.
And finally, there is the aspect gorilla mentions: do you want some ready to use, "survival" expression or ideas about how it can be asked "properly".
Without knowing all that, I would say a) you've found a simple but correct translation for the English sentence (first formal, second informal) but b) translating the English word by word does not express whether you want just a free bag (if they have any) or you would be ready to buy one if it was possible (because you need one).
In either case, you could say:
Kérek egy táskát (/szatyrot). - I would like a bag/carrier bag (please). (Fairly short and direct but easy.) or
Kaphatnék egy táskát (/szatyrot)? - Could I have a bag/carrier bag (please)? (More polite.)
Thanks for your responses –– I was mainly wondering as to whether the 3rd person possesive is used when addessing a 2nd person subject in order to be polite.
When it comes to the actual question, I'm mainly trying to ask if there are free plastic bags available, and I figure that asking this is tantamount to asking for one.
I also just generally need to know, when entering a shop, how to ask if they have whatever it is I'm looking to buy. It seems that using the verb kapható is a good way to do this. I have not gotten to the adessive case yet, but I will return to it when I do––
The closest and best literal translation is gorilla's version: Táska kapható? Kapható itt táska?
It would be possible to say it more colloquially: Táskák vannak? or Táska? Van?
Zsanna's versions mean you expect they sell bags.