There were extra hot dogs. This is where I have a role to play; to help eat leftovers.
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Yes, semicolons should separate complete clauses, if a period is too strong of a separation; use it rarely (more rarely than I am; adverbial phrases are preferred). Your sentence deserves a colon, since the final phrase is an example of your role to play; it might have been a list.
The correct usage of a semicolon is to separate to related independent clauses.
There were extra hot dogs. This is where I have a role to play; I help eat leftovers.
Or, if you still want to have a dependent clause at the end, use a colon instead.
There were extra hot dogs. This is where I have a role to play: help eat leftovers.