Absolutely. It is actually in the bylaws of the town.
Please.

What I was trying to get at is that some customs are regional. We have seen that here on the boards when discussing different wedding customs. Some regions think the money dance is tacky, some believe it is wonderful.
It is the same for birthday parties. There are regional customs. In our "circle", food is always served. Usually it is pizza. The kids take a bite, go play, come back for another bite, go play, etc.
The OP should go with what is customarily done for parties in her area and her circle of friends. If cake and ice cream is the norm - go for it. If food is normally served, then parents might expect it.
Having been to more birthday parties in the past 10 years than I want to remember, I can't remember one without food. So, in OUR region, food is the norm. Parents come to expect it. We may be all Stepford wives, but we all do it.
But it is completely possible that in other areas, parents would be surprised to see food at a party.
OP - do what you are comfortable with. It IS your party.