I have a lantana and a ground cover that I want to remove from my rosebed. I have tried just digging them up, but the root system is such that they just keep coming back each year. Is there a safe, organic method to use to remove these plants?

Yes, but I wonder why it is that you wish to remove the lantana, a beautiful native plant. What kind of groundcover is it? It may be best to leave it there unless you would rather have mulch around your roses.
You can spot spray with the following mixture:
20% or 10% vinegar per gallon, add 1 oz orange oil
and a teaspoon or so of Dr. Bronner's peppermint soap or some other kind of mild soap.
This will kill anything that you spray it on except bermuda grass. This persistant turf will just turn brown then come back.