Sometimes they can’t get you on a new flight right away. You may have to do an overnight stay if there are no available flights. But with all the cancellations going on right now, that’s pretty much a possibility on any flight these days no matter where you’re going. And even if they rebook you for the same day, there’s no guarantee your luggage will make the new flight. It may end up at the original destination or be left behind if the transfer is tight. Make sure your carry on has stuff for an overnight stay, like a change of clothes, toiletries, medication, etc.
Emirates is a partner airline of JetBlue, so had the OP bought both tickets via the JetBlue app or website, Jetblue would be responsible if the JetBlue flight is late enough that he misses the connection to Emirates. But in this case, the ticket was purchased through Travelocity, so I don’t know if the JetBlue protection applies.
Jetblue might be financially responsible, but if your next leg is on an Emirates aircraft, I think they would coordinate the process of looking after you - which might get you better treatment. I don’t think it makes any difference that the ticket was purchased through an agency.