Is it only possible to do one of the quests offered by Vex and Delvin? I did one each for them and now they won't talk to me about their other quests.

you can only do one at a time from each person, if thats what you mean.

After you complete their quest, you should be able to pick "I'm ready for another job" or something like that.

^^ After you complete the assigned job, sweep/numbers for example go back and tell them you have completed it ... obtain your reward. Tell them you are ready for more work ....select new job

n.b. make sure you see on screen that you have accepted the quest and the details of the job are shown BEFORE you talk to the next NPC for their job.

I made this mistake last night and it took me a while to work out that I was talking to the other person too quickly and not allowing the first quest to register in my journal (PS3)