It's a simple metaphor. A barrel is something that you store things in. So to "reach into a barrel" to find something is a way of saying, retrieve something that was stored.
A barrel is not a particularly good way to store things, as everything is just piled on top of each other and jumbled together. We usually store things on shelves or in closets or in trunks. A barrel is generally seen as a rather careless way to store things. Items that you don't know what to do with, you throw in a barrel. So if you were looking for some odd item, you might search for it in a barrel where you're keeping miscellaneous things. So if someone is "reaching into a barrel" to get something, that rather implies that they're looking for some random item or something not particularly valuable.
That fits this case. They couldn't get a "good" speaker, so they had to grab the first available speaker they could find. They just reached into a barrel and pulled out whatever they could get.
"Notoriety" means fame, in a negative sense. We say that someone is notorious for his scandalous or criminal behavior.
A common idiom is to refer to someone's "claim to fame", that is, what he did that made him famous. Like, "Fred's only claim to fame is that he had a minor role in a movie 10 years ago." I think "claim to notoriety" is a play on this. What is it that made him notorious?
The phrase "claim to notoriety" by itself doesn't mean that the person is proud of it. It's not that he's claiming it, just that it's ... there. In context he might be proud of it.