What Authority does an O-3 have in regards to writing Certificates of Achievement (not citation)?

AR 600-8-22, 5 March 2019, 10-7. Acts and achievements that don't meet standards for decorations, as local commander may prescribe.
One question for consideration is what value this has for the recipient. Admittedly, I'm not current with regard to promotion point value that LTC and above may have, as it did during my period of service. Different Soldiers will value a certificate signed by a CPT differently. Some may treasure it, while some may prefer a 3-4 day pass. A written command policy that includes a pass for someone so recognized would likely increase worth.
Good to see Commanders still looking to take care of Soldiers.

I am not sure how promotion points work for Certificates of Achievement now; however, when I was in we always pushed them up for LTC or above signature since that was required for the Soldier to get promotion points. If promotion points are not a concern, you may award as a Commander.

You can right them sir, but like others have stated it isn't worth promotion points unless signed by an O-5 or above.