You are quite right!
Many local web sites here are glossy but in terms of user interaction and useful information presentation they are the pits.
http://autodromoalgarve.com.pt is no exception.
I don't know of any agents who do tickets. There seem to be some indications on the web to the fact that FNAC (Portugal only?), Media Markt and El Corte Ingles may sell tickets for events but its not clear.
I suggest that you email or call this person whose details are shown on the contacts page:
Members & Tickets’ Information
Name: Catarina Azevedo
Email: catarinaazevedo@parkalgar.com
Phone: (+351) 925446084
I would write/call in English. Most folk down here in that type of position speak at least some english.