In TOS era, no. The original series was going to have shuttlecraft only, and so we saw shuttles OR transporters, but not both. TNG saw shuttles evolve to become more like ships in that regard. TNG era (Type 7 shuttles) had the shuttle escape transporter
The shuttle escape transporter was a small short-range personnel transporter used aboard Federation shuttles. This transporter was manually controlled from a control console set in the ceiling of the passenger compartment of a shuttle.
In official canon, that's the first time we see a shuttle use its transporter.
How did Spock's shuttle get one? Just because they weren't previously depicted doesn't mean they didn't have one
The best take is to assume that in-universe, the technology is the same, but the original series lacked the budget to portray it as it "really was". Look at the difference in the tech between the pilot and the first episode - the "lasers" became phasers, and the communicators got a lot more sleek. An in-universe reason for that isn't necessarily needed - you can imagine that the design from TOS is what they used in the pilot, and that the look of the tech in Discovery is what the tech should look like in TOS.