Hello,
I had to replace a transformer in a Carrier handler.

The transformer is a honeywell universal transformer with four wires on the primary side and 2 terminals marked R and C on the secondary side.

My tech won't be able to arrive until next week so I wanted to get the unit running for now. I bypassed the thermostat for now and was able to get the handler running but still unable to get the condenser running.

According to the schematic it appears that the black/orange wires get hooked up to the lead for a 240 connection and I taped off the remaining red and white individually.
I then connected the r and g wires from the handler to R terminal on the transformer
And the c wire from the handler to the C terminal on the transformer.

That kicked the handler on but I can't seem to get the condenser running.

I have 2 low voltage wires coming in from the condenser to the handler red and white.

How can I jump the condenser for now and get it running until the tech arrives?

Is there a way I can jump it directly from the condenser unit itself?

Leave air handler wiring alone, or put back how it was, can do from thermostat wiring.
Yellow or blue may be for cooling,
Green for Fan.
White for heat.
What color wires at the thermostat?
What colors do compressor wires hook too?

If wanting to do at air handler, hook the pair from compressor to Transformer secondary.
Outside Compressor Contactor needs 24 Volts AC to Energize.

I heard some humming/buzzing from the condenser.
Pulled cover off and see a spring loaded switch by the power leads. Gave it a little tap and the condenser came on.

So right now I have the white wire from the condenser connected R on transformer
And red wire from condenser connected to C on transformer

At the condenser the side:
The red connects to an orange wire which connects to something that looks like a circuit board or relay.
And the white wire connects to a blue wire which leads into the condenser.

I was originally under the impression that
Red should connect to R on the transformer
White should connect to C on the transformer

Should I try to switch?
Will it harm anything either way?

Your transformers common connects to the relays/contactors coil for Blower, Heat, Compressor Contactor.
At the thermostat when you set to cool, it takes the 24 volts(red) and connects to green for fan relay AND Compressor Contactor(Maybe blue or yellow), and for heat the thermostat connects the red to the green and blue or yellow, so inside blower and compressor turn on together.