Can allergies cause a sore throat?

They certainly can. Allergy symptoms very widely between different people. They can include itchy eyes/nose/mouth, running eyes/nose, sore eyes/throat, and many more. Antihistamines usually work well to alleviate some of the symptoms and can be taken as one tablet every 12 hrs or 1 tablet every 24hrs. Have a chat to your local pharmacist to see what they recommend. It's very unfortunate that you are allergic to your pets. Perhaps limiting their access to parts of the house (ie; NOT allowed in the bedroom) and regularly washing their bedding, plus not permitting them to get on the couch might help. Frequent hand washing (especially before touching your face) will also help. It's such a shame that you are allergic to man's best friend (or 3 of them). Best of luck.