What is the pronoun Do you have any room left in your storage shed for my lawnmower?

There are two pronouns in the sentence: your and my.

The pronouns 'your' and 'my' are both possessive adjectives, a Word placed before a noun to describe that noun as belonging to someone or something.

The possessive adjectives are: my, your, his, hers, its, our, their

You will note that, unlike possessive nouns, possessive adjectives do NOT use an apostrophe to show possession.