Join a research group at a university I do not attend I work as a data scientist and many of the problems I am presented with require a non-trivial amount of research to solve, and so, a portion of my work week consist of reading relevant literature.  

I throughly enjoy doing this, and in doing so, wonder if there is a contribution I could make to a local research group.  More specifically, would it be completely out of place to write a local university and ask to join a research group or is this something that is frowned upon?  

In my head I envision this as an interesting symbiotic partnership where students could be exposed to methods from the private sector and I get to scratch my academic itch, but I certainly don’t want to step on any toes or have the appearance of taking someone’s place.



EDIT:

By "join a research group" I mean finding a lab that is working on a problem I am interested in, going to said lab, meet the members, help solve a problem in anyway I can (writing code, discussing ideas, etc...).

Also for context, I am in the US.
This will depend on the field. In my area (in computer science), research groups frequently work with external collaborators from industry. From the perspective of a research group, such collaborations can be very desirable, since practical motivations or, even better, applications can improve their research. There are even conferences with a specialized "industry" track, where perspectives of practitioners are sought explicitly.

Teaching-wise, this will depend on the country as well. In Germany, it's not unheard of that people from industry are employed on a part-time contract to give a lecture with a practical focus.

I think you should directly contact a suitable research group (professors and maybe post-docs), rather than the university, since employees of the university administration might be unaware of those specifics.