I often get annoyed by empires that start out ignoring me, then as soon as I start growing and border friction increases they start insulting me non-stop, and then at some point (possibly due to me getting a better diplomatic rep through science as well as being more powerful) they suddenly start begging to be my vassal. After all those insults that I could do absolutely nothing against, I just want to reduce them to dust at that point. They had their chance, don't come crawling now.
But to answer the question, I get at least one empire asking to be my vassal every game. As stated above, it happens more often during the end-game crisis if they just happen to exist in the affected part of the galaxy. Although at that point the only thing I could possibly do for them is to allow them to settle a world in my empire if that were an option. By the time they ask, they've already lost half their empire and there's nothing I can do. The open borders are nice though.
Also, I recently had a criminal empire right next to me. After being forced to kick their branch office off my capital about a dozen times, they started asking to be my vassal. Firstly, I wonder if it's really appropriate for a criminal empire to be anyone's vassal, much like Marauder Empires they should just stay unaffiliated. Second, I wonder if a criminal empire vassal is disallowed from building branch offices in their master's empire. It would be truly annoying if they started building criminal branch offices all over my worlds and all I could do is declare war on my vassal. I think it's related to opinion though, so as long as they like you, they'll refrain from building in your worlds.
Completely unrelated, but how often does the end-game crisis actually ever win? I've played many games now where I just sit there, guarding my wormholes and the Preths never even make it to my borders. They make it through one, maybe two empires, then they get squashed by the Fallen empires. It's really boring really. I would love just strategically sacrificing my worlds while retreating to the L-Cluster with its single point of access and just let the end-game crisis take over the galaxy.