I've finished Vex's and Delvin's side jobs for Windhelm, Markarth, and Whiterun. The only city I have to do quests for is Solitude to get the achievement, but neither Vex nor Delvin will give me jobs in Solitude. Is this a glitch, or do I have to get Solitude quests from another person?

I have the same glitch only with me I can't get the last City job in Riverrun. I knew something was up when Delvin would not accept the Larcency items (the Map) and provide payment. I can take quests but they just go on forever.
Hopefully they fix.

I have the same problem. I read somewhere that you need to keep doing bedlam jobs until you've done one in every city, but the guy who gives them out never changes things up.