Can the comma next to “bleeding” be placed next to “and”?

Yes:

The platelets are blood cells developed in the bone marrow to form clots to stop or prevent bleeding and, without them, the blood vessels can’t heal.

Or none at all:

The platelets are blood cells developed in the bone marrow to form clots to stop or prevent bleeding and without them the blood vessels can’t heal.

Or use two sentences:

The platelets are blood cells developed in the bone marrow to form clots to stop or prevent bleeding. The blood vessels can't heal without them.

I would say:

The platelets are blood cells developed in bone marrow to form clots that stop or prevent bleeding; without them the blood vessels can't heal.