I'd personally use 'can I have' and 'can I get' to ask for food, because these can be made into interrogatives.
Can I have the pie please?
Can I get some chips too?
You can also use 'I'll have', but this is more of a statement.
Example: I'll have the chicken please.
I'd avoid 'let me get', because it may be seen as too direct (in Britain anyway)!