Hi Autumnleaf!
I actually do microdermabrasion and a light chemical peel between my LHR treatments.
I've done a lot of research and the general consensus is to wait at least 10 days. Some even say two weeks, some four weeks. I've had three microderm sessions already that were spaced in between my LHR. My LHR sessions are roughly 5 weeks apart.
You need to give your facial skin time to heal before you go about seeking the other treatment since I'm sure you do not want to damage your skin.
This whole time in between actually depends on the strength setting of the laser as well as the strength of the suction they set the microdermabrasion machine at, so it would be best to ask the techs/doctor (at both) to see how long you should wait.
No need to be embarrassed about asking them about it. If necessary, just say you want to get rid of acne scars (when asking the LHR tech) or say that it'll just be easier for you to clean shave every day for work (when asking your doc after microderm).
I was embarrassed during my first appointments too but I soon realized that there was absolutely nothing to be embarrassed about. I walk in there like a BOSS every time
Walk in there like a boss with your head held high and ask them about the waiting time between sessions. You can do it.