Once I asked in Prague about a hotel "Szukam Hotel Villa" and I was lectured that I should learn that I'm not in Poland any more:D
I would say the latter is the case here ;) This sentence suggests that she is looking for a husband, because she doesn't have one. If you want to say "She is looking for her husband" it would be "Ona szuka swojego męża".
Swój is "one's own" in English. Very few people use the pronoun 'one' in English, but it helps to remember. Ona szuka jej męża means "She is looking for her (someone else's) husband"
Swój is used because the person which the possesive is referring to is the same as the subject. I found some explanation, hope it helps: