Believe it or not, and I know I was surprised, there is a Log Cabin restaurant in Grand Rapids, Michigan. I don't know if it's the same one as in the book, but the real one has been there since 1938. And just like there was a band in the Log Cabin when Bud got there, the Log Cabin today still has concerts. From what I could figure out, Bud, Not Buddy takes place in 1936, but maybe Christopher Paul Curtis based the Log Cabin in his book on the one in real life. Here it is:
Now imagine a Packard like this one parked out front.
Unfortunately, after a lot of searching, I couldn't find a restaurant called The Sweet Pea.