The closer synonym for the context is to allow, so it would be like:
[...] provide the following details in order to allow us to provide you a proper service.
But, although it's a synonym of to enable, it has a slightly different meaning.
- To allow someone has the acception of
"to permit someone to do something/to give permission".
- To enable is similar, but it has the acception of
"make it possible/facilitate".
In the end, I think your choice was the best one.
Here are the other synonyms, in case you want to have a wider view, although I think none seems specifically fit for the given context:
Allow, permit, let, give the means, equip, empower, make able, fit; make possible, facilitate; authorize, entitle, qualify.