Billund isn't so big, but as the home of LegoLand, it has a lot of tourist stuff.

Here is a hotel that rents bikes for 50 crowns a day (but you probably have to stay there at least one night, I suppose):

http://www.legoland-travel.dk/pakke-...hotel-billund/

Fricykler is a popular bike store in Denmark. No idea if the rent bikes, but they will know someone who does -- try them first:

http://www.fricykler.dk/page.asp?sideid=320&zcs=209

Fri Cykler - Billund
Cykellageret Billund ApS
Granvej 2A
7190 Billund
Tlf. 75331203
cykellageret@mail.dk

Here is a local bike club in Billund. They can probably put you in touch with someone:

http://www.bicm.dk/

Otherwise, there are plenty of places in Vejle that will rent bikes. You can take a bus from the airport to the train station, and the train to Vejle. Keywords for searching:

bike = cykel
to rent = at leje
bike shop = cykelhandel

Have fun!