There are many companies offering shuttle services between Szeged and the Airport. You can find one by browsing "Airport-Szeged shuttle" in google. Below we give you a few suggestions.
You can also check out the country's most popular carpool platform, where you can find both individuals and companies sharing available seats in their vehicle. Unfortunately the site doesn't have an English version, but you can translate it via your browser. It is worth the trouble because there are many reliable and cheap service providers listed here, just make sure to find a highly rated one.