You should be able to do what you describe. The large vertical pipe looks to be a waste stack. The smaller vertical pipe looks to be a vent. The washer discharge is counted as 2 drainage fixture units and requires a 2" minimum drain, which it looks to be. A laundry sink is also counted as 2 DFU's and requires a 1-1/2" minimum drain. Your 2" drain can handle up to 6 DFU's, so you should be able to add the laundry sink drain to the 2" laundry drain pipe with no issues. I would cut the laundry p-trap off flush with its' elbow and leave a projecting pipe stub in the adjacent vent tee, then add a sanitary tee or wye for the new laundry tub, and then a new p-trap for the washer discharge. Assuming your laundry sink will be close to the washer, your existing vent location will be fine (at 1/4" slope per foot, it can be up to 8' feet away).

FWIW, I prefer a wall mount laundry sink, no feet = no obstructions to collect crud/debris, much easier to sweep around. One possible manufacturer:

https://mustee.com/products/laundry-tubs-and-sinks/laundry-and-utility-tubs/

The box stores usually don't carry the wall mount sinks and must be special ordered:

https://www.homedepot.com/p/Mustee-23-in-x-23-in-Thermplastic-Wall-Mount-Laundry-Tub-15W/202822596?source=shoppingads&locale=en-US&mtc=Shopping-B-F_D29B-G-D29B-29_23_TUBS_AND_SHOWERS-Multi-NA-NA-PLA-NA-NA-NA&cm_mmc=Shopping-B-F_D29B-G-D29B-29_23_TUBS_AND_SHOWERS-Multi-NA-NA-PLA-NA-NA-NA-71700000032906557-58700003861657818-92700063020206436&gclid=EAIaIQobChMIoeLzotq_8QIVT__jBx2h1Au8EAQYAiABEgL-dvD_BwE&gclsrc=aw.ds

Thank you for the response! Sorry if it’s asking too much, but would you be able to make a crude drawing of what it is you’re describing? Just to make sure I am understand the setup correctly.
And yes, I have already bought a wall mounted sink by mustee that will be mounted just to the right of the washer drain now.
Thanks again

From left: Stack tee - vent tee- sanitary tee for sink - new p-trap for washer

Like this?