I live in a 7 year old house I moved into when it was new. I have two floors on the main house and a full basement. I have two full baths and a utility room on the top floor. I have a half bath on the main floor of the house and a full bath in the basement. Recently when the washer is draining I have air gurgling back through the toilet in the half bath on the main floor. Do I have a vent pipe stopped up on the roof?

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Yes and no, your vent is stopped up but not at the roof. You will be rodding from the roof but the blockage is anywhere along the vent, most freequently at the bottom. Rod with a sewer rodder roof to basement floor plus 15'

It sounds like air way to the stench pipe may be blocked if pipe terminates on roof make sure bird terminal is still on could have bird nest in pipe if pipe does not terminate on roof or outside you may have a durgo valve fitted which may be faulty-as advised check that there is no blocked pipes

The stench pipe is the air way, also called the vent stack. Some have caps but most are just open to the air and that's how they can get clogged.