There is no direct bus from Seville Airport to either Ayamonte or Vila Real Stª. Antonio.
There is a regular bus service from Seville Airport to Seville stopping at Estacion Sta. Justa and El Prado Bus Station. From El Prado you would have to get another local bus to the Estacion das Armas Bus Station.
Mondays to Saturdays the first bus leaves the airport at: 05:45 Sundays at 06:45.
There are buses every 30 minutes until after midnight, 23:45 on Sundays.
Timetable here:
http://www.tussam.es/index.php?id=200
From El Prado to Estacion das Armas I would recommend the C4 Bus in which runs from just after 6 in the morning till 23:30.
From there there are twice daily services to Vila Real Stª. Antonio. operated by Damas/Eva. See: http://www.eva-bus.com
The current Seville to Vila Real Stª. Antonio timetable is as follows:
Seville 07:30 Vila Real S António 08:55 Daily Portuguese Time
Sevilha 16:15 Vila Real S António 17:40 Daily Portuguese Time
Damas also operates several services a day to Ayamonte from this bus station. You can find their site here:
http://www.damas-sa.es/
The current Seville to Ayamonte timetable is as follows:
Monday to Saturdays (Work Days) - 09:30
Saturdays, Sundays and Holidays - 11:00
Mondays to Fridays - 11:30
Mondays to Fridays - 12:30
Daily - 15:30
Sundays and Public Holidays - 19:00
Monday to Saturdays (Work Days) - 19:30
Transit Time takes about 2 hours.
Finally, be aware that taxis are expensive from the airport into Seville.
