Are there any good online MBA programs that don’t require any letters of recommendation or only require contact information of references? E.g. Wake Forest and Nebraska only contact references if they can not make a decision based on other application materials provided.

[Edited by Bk5522 on Mar 25, 2022]

A letter of recommendation strengthens your application. It is a bit strange to try to avoid providing one. You should have at least one or two people you have worked with that can speak highly of your work and your integrity. 
Personally, I think this should be the last criteria for choosing a business school for your MBA.