Learn Spanish (Latin America): Yo mismo aspiro el colchón.

Adding Yo mismo emphasizes that you are doing the action yourself or personally, rather than having someone else do it. Without it, the sentence just states the plain fact: I vacuum the mattress.

Would a female speaker still use Yo mismo?

No, a woman would say Yo misma. When mismo is used for emphasis with a pronoun, it acts like an adjective and must match the gender and number of the person it refers to.

If I want to say 'I vacuum the mattress itself,' where do I put mismo?

You would place it right after the noun you want to emphasize: Aspiro el colchón mismo. Because our sentence places it directly after Yo, it emphasizes the person doing the action, not the object being vacuumed.

I learned that aspirar means 'to inhale' or 'to aspire'. Is it the normal word for vacuuming?

Yes! In Latin America, aspirar is the standard verb for vacuuming (using an aspiradora, which is a vacuum cleaner). The context of cleaning a mattress makes the meaning obvious here.