My right foot is swollen for no reason? No trauma. Could it be infected?

Possibly. While infection may create inflamation with associated swelling, unilateral leg or lower extremity swelling is also associated with other conditions including a DVT and should be evaluated promptly.
Just one side? Unilateral swelling without trauma could suggest venous insufficiency or thrombophlebitis. A venous doppler or ultrasound can help make the diagnosis. If there is redness and warmth especially around the great toe joint this could an acute gouty attack. Infection is less likely especially w/o a skin ulceration or open wound. See a doctor.
It could be.... Some causes of swelling can include systemic problems such as hypertension, and problems with the kidneys and localized issues such as arthritis, infection, lymphatic obstruction, blood clots, varicose veins, trauma (fracture, sprain, and tendonitis) and side effects to medication. Have it evaluated by a professional and get the appropriate treatment.