What are native ground cover plants for zone 6a for shade and also some for sun that can be used in flowerbeds? 

Thank you for your question to Ask a Master Gardener Volunteer with eXtension.

Let me first apologize for the delayed reply.  I just received your question yesterday afternoon.  Not sure what happened on our end to warrant this delay.

You are being very conscientious when it comes to the use of natives as a ground cover material, I applaud you!

I would advise you to take a look at the following sites, all have great information about the variety of native plants that can flourish in Ohio:http://www.plantnative.org/rpl-inoh.htm

https://static1.squarespace.com/static/5d5cad7c0df7c<personal data hidden>/t/5e5a8343bcd48e5c8c20019f/<personal data hidden>94/Bob+Henns%2C+Debra+Knape%2C+Hope+Taft.pdf

The first site is very user friendly and lists many common native plants and their heights (check under the perennial area).  The second site is very extensive and you will be surprised how many options do exist (I know I was!).

The following two sites give very good basic information about the use of native plants:

https://www.bbg.org/gardening/article/native_groundcovers

https://choosenatives.org/articles/plant-native-ground-covers-make-america-green/

Good luck with your quest to go native!