The router itself only has a 1/4'“ option.
I believe that even if you could find a 1/4” to 3/8” adapter, it would not be recommended. (It sounds quite dangerous to be honest. Because you want something larger than the router, it would need a mechanism to hold it closed in a way that is perfectly balanced because you’re spinning it at up to 30k RPM and strong enough to hold tools being driven into other materials.)
Depending on how keen you are, you could replace the Makita router with a completely different one, fitting it into housing tube the same diameter as the Makita. Keep in mind that to find a compact, 3/8” drive router you may have to consider a water-cooling with the complexity that entails.
The boxzy is a great machine, but when you start wanting to work on larger pieces and with larger tools, it might be time to consider building a bigger machine. I can’t work out how to drive a 4th axis in the confines of the very compact chassis so I’m using the boxzy (and an awful lot of t-slot extrusion) to build a bigger machine.