This is a passive construction.

Je voudrais vous faire une radiographie. C’est moi qui vais vous la faire.

Je voudrais vous faire faire une radiographie. C’est une troisième personne qui va la faire.

As in English with the construction make sb do sth.

The construction "faire faire" means externaliser, sous-traiter, concéder (un travail, un marché) à un sous-traitant.

See here https://www.anglaisfacile.com/exercices/exercice-anglais-2/exercice-anglais-105563.php

for a nice explanation.

Also here (in French): https://lemondedufrancais.com/2012/02/17/construction-passive-se-faire-infinitif/

Dimitris’ answer is correct; a little more about the two uses of ‘faire’ in this sentence, because the confusion comes in part from those two uses in the same sentence.

Faire has the obvious meaning "to do, to make", as in

Je vous fais un cafe? (Shall I make you a coffee?), or

Je voudrais vous faire une radiographie. (I would like to do your X-Ray).

The second use of faire is the passive construct:

Je vais me faire couper les cheveux. (I’m going to get my hair cut)

Je vais vous faire faire une radiographie (I’m going to get your X-Ray done|to get someone to do your X-Ray)

So in faire faire, both senses of the verb as used in succession, which is often translated as "to get done".