All you can do is cut, pull or spray.

What do I spray with? Weed killer or something stronger?

Are you sure it's sycamore and not maple? I only ask because we get little maple trees popping up everywhere from the helicopter seeds that fly from the maples.

Use brush killer on the sprouts.

However since they are probably seedlings once you pull them from the ground, that plant will not come back, as it does not have a large root system. The only way the brush killer would work is if you use a paint brush to put it on the cut ends of the seedling stump. If the seedlings are among your flowers and plants, you can not apply the brush killer as a spray as it will kill everything including you plantings that you have worked hard to make nice.

For those in the yard just cut them with the lawn mower.

You could not use brush killer in the lawn as it would be harmful to the lawn.

This sounds like a situation I've run into in Florida with live oak .... the tree is cut down and a billion sprouts emerge from the roots, which are not yet dead. (It probably happens with all kinds of species if they have the opportunity.) They take 3 or 4 years to die. Meanwhile, sprouts continually emerge from the roots in their desperate attempt to stay alive. In lawn, the regular mowing of grass will finally deplete the roots of energy. In beds, they must be manually cut or sprayed with glyphosate, 2,4-D or similar herbicide (not brush killer), being careful not to spray "good" plants.