It is possible to produce aspirin by reacting salicylic acid with acetic acid. This reaction is done in the presence of an acid catalyst and produces an ester.
Although, the reaction with acetic acid is less favourable. In excess water and presence of an acid catalyst, the reaction will go in the reverse direction. That is hydrolysis of the ester. This will cause a significant decrease in your yield unless you set up an apparatus that removes the water or the aspirin as soon as it is produced.
So unless you have a particular issue with acetic anhydride this probably is not an easy way to get a good yield of aspirin.
It is possible to produce aspirin by reacting salicylic acid with acetic acid & sodium acetate by heating one and half hr. & pouring the reaction mixture in ice-water mixture to get solid aspirin.
I think the obvious answer here would be the use of acetyl chloride with some DMAP and base. Alternatively, a Bayer-Villiger reaction on the corresponding arylketoacid/ester may be a good option.
Reacting salicylic acid with acetyl chloride would give you aspirin (acetylsalicylic acid), but would yield hydrogen chloride gas as a by-product instead of acetic acid as the by-product (which you would get from using acetic anhydride as the acetylating agent rather than acetyl chloride)