Yes, ~50% of DO grads for ACGME residency and get acredit that way.
Most do USMLEs and the COMLEX, some do just COMLEX (some ACGME residencies will except the COMLEX). DO's apply through the same MATCH as MD do and in most places get same treatment during the complete application process. (DOs also have an AOA match as well).
There are joint AOA/ACGME residencies as well, but those are few and far inbetween. YOu can then get accredit be allopathic board of _____ (specialty).