[WW2]
The Germans and the Americans developed their half-tracked APCs(Sd Kfz 251 and M3/M5, respectively) in order to fit in with their Blitzkrieg theory of war, as in, carrying troops to the front line quickly so they can finish the job their panzers missed. I know the Soviets had a pretty similar war theory called "Deep Battle"-but if that was the case, why did they lack development in vehicles, like said half tracks, that could quickly carry troops to the front? Because as far as I know the Soviets used Lend-Lease M3, Stude 6x6s, and Willys MBs to do the jobs instead of making one of their own APCs/Troop Transport vehicles.
I'm going to break your question up into two parts and answer them separately.
Why didn't the USSR develop halftracks?
They did. The Scientific Automotive Tractor Institute (NATI) designed the GAZ-60 halftrack based on the GAZ-AA truck and ZIS-22 halftrack based on the ZIS-5 truck. These vehicles showed improved off-road mobility, but greater fuel consumption and lower speed than regular trucks. Research focused on all wheel drive trucks since that looked like a more promising project. Instead of halftracks the Soviet government developed fully tracked tractors. The Komsomolets tractor, accepted into service in 1939, was arguably more versatile than the Sd.Kfz.251 since it could be used to tow artillery, transport goods or infantry, or in the worst scenario as an ersatz tankette. It also served as a base for the ZIS-30 tank destroyer.
Would APCs let the Red Army quickly carry troops to the front to carry out the Deep Battle doctrine?
Deep Battle was not about getting troops to the front quickly. It was about exploiting a breach in enemy defenses by conventional means by inserting a rapidly moving exploitation force. The exploitation force engaged rear echelon troops, front line troops that were withdrawn and recovering, ill-prepared units, etc. These units would not be fully set up to meet the Soviet force, thus moving quickly was more important than protection. Moving infantry via trucks or other unarmoured transport was considered fine for this purpose.
Sources:
http://www.tankarchives.ca/2017/02/tank-corps-in-attack.html
http://www.tankarchives.ca/2017/04/halftrack-experiments.html
http://www.tankarchives.ca/2017/02/self-propelled-artillery-on-tractor.html
http://www.tankarchives.ca/2017/02/htz-16-improvisation-on-industrial-scale.html