I like chocolate, vanilla, and lemon and orange ice cream.
Indicating "lemon and orange" is a combined flavor, as an item in the list needing an initial and.
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You can repeat and as often as you like if you think it conveys your meaning. There is no rule of English grammar that limits its use. The placing of a comma after vanilla makes all the difference. Without it, the sentence leaves open the possibility that your preference is not for lemon and orange ice cream, but for vanilla and lemon ice cream.
The sentence is correct as written and does indicate that lemon and orange is to be treated as unit comparable to chocolate. However it may be unclear to some readers, and resists being read quickly.
You might want to consider using lemon-orange to describe the flavor. It is unabiguous and scans quickly and easily.
You certainly can use "and" more than once in a sentence, but in this case I would consider rewording it somehow instead. I would definitely not use an ampersand. Actually, I think removing the second "and" would make the meaning much clearer. Try: "I like chocolate, vanilla and lemon-orange ice cream."