No doubt the Avery software either does not work with 2007 or they
simply hardcoded which versions they look for in setup and it does not
yet include a check for 2007.

Contacting Avery would probably be the best course - but if it's just
a template, you might try unpacking it and getting to the actual
template file on a previous version of office and then just shuttle it
over to the 2007 machine.

There's nothing that a template from Avery can do that cannot be done with
mail merge in Word.  Just select the type of label that you want to use in
Word.