They don't seem to offer season tickets to Luton Airport, presumably because Luton Airport isn't a station (whereas Luton Airport Parkway is) and to get to Luton Airport itself you would require a bus.
BR Fares is an excellent website that demonstrates the fares on offer. For
Bedford to Luton Airport, there are indeed no season tickets, just a selection of walk-up fares. For
Bedford to Luton Airport Parkway, there are indeed season tickets. (A quick search at a few others suggests that Bedford is not alone).
Presumably the logic is that most people going to the airport are traveling on flights, and therefore not regular travellers (although this would obviously ignore the large number of people who commute to Luton Airport to work, like yourself. Perhaps the bus operator doesn't want to accept railway season tickets? Perhaps it's just an oversight, in which case writing to Thameslink may* yield results.
So I think your best option would be to buy a ticket to Luton Airport Parkway, and pay the bus fare separately (£3.40 cash only, which is quite pricy and a bit of a pain). I can't see a season ticket for the bus, and sadly the Luton PLUSBUS specifically excludes the shuttle to the airport. Perhaps, since the bus is operated for the airport, your employer may be able to help? A quick search reveals the
Luton Airport Staff Discount Travelcard, which may be of use to you. Of note, apparently you can get a 50% discount on the cost of your season ticket! I'm sure your employer can give you more information if you are eligible.
*I suspect not, but an e-mail doesn't cost much.
