if the goy goes on his own and purchases something on Shabbos which is missing from the party can you use that item at the party
Essentially since the people he is buying the items for are Jewish, it is prohibited, however might be room for leniency under certain conditions. If the gentile had to a melacha d’orayso in order to buy the item, (he needed to drive there, or he brought the item thru a reshus horabim) then we cannot benefit from what was purchased. If however the gentile would didn’t have to violate a d’orayso, (he didn’t have reasonably have to drive there, and he didn’t take it thru a resus horabim, etc.) then it would depend on how important the item that he purchased is. If it is only an extra, but they can manage without it, then it is not permitted to benefit from what he brought. However if the item that he brought was essential for the Shabbos meal, i.e. they forgot to buy cups to drink with, then they may benefit from what was bought. The reason for this is because there is a heter called shevus d’shevus b’makom mitzva. The gentile doing work for the Jew is a rabbinic prohibition, and if the gentile only violated a rabbinic prohibition, (called shevus d’shevus) and it was essential for the Shabbos meal, (b’makom mitzva) then it is permitted.
As a side point, I would recommend that you buy the sefer “The Sanctity of Shabbos” By R’ Simcha Bunim Cohen, which goes thru these halachos.
All the best