I've got asthma, allergic rhinitis and multiple food allergies. Could it be possible that I'm also allergic to my husband's sperm? Is there a test?

Unlikely allergic. It is very unlikely that you are "allergic" to his sperm. Men can develop antibodies to their own sperm ( usually after vasectomy reversals) but it is very unlikely that you would develop these antibodies.
Allergy to sperm. Yes, you can be allergic to sperm and also to latex condoms. Discuss this with your doctor.