It is probably easiest to do what you say in your last statement: Run a new 120V cable and control it from a new relay, controlled by the same thermostat signal as the blower.
If there is an unused neutral in the cable feeding the blower (make sure it's connected at the panel), you could possibly use that to power the new gadgets using either of the live wires , but you still need a new relay. The reason for that is the existing relay will disconnect the two live legs from each other but will not disconnect either of them from its source, so there is nowhere you can tap off of the existing relay that is "off" in relation to your neutral.
If you do run a new 120V cable, it doesn't have to be from the panel, it could be from the nearest available junction box on an existing circuit. But you should install a manual cutoff switch outside the blower housing, next to the blower's one, to make it clear to people doing maintenance that both must be shut off before servicing. It would be safer to use one of the existing live legs but if you don't have a neutral that would require replacing the existing feed cable with a /3 one.
I may be wrong, I'm making assumptions. If you upload the wiring diagram for your blower and add it to the question and a picture of the relay, that would help to confirm this.