Installing nema 14-50 with no ground
Nope, no good.
The right thing to do in this situation is not to just omit the equipment ground wire from the four-wire receptacle as if it was optional.  It will operate, but it will not be code compliant, and it won't be as safe as it could or should be.  That wire is after all the safety ground.
If you aren't prepared to run a ground wire for the circuit, the right thing to do in this situation is use the three-wire receptacle, and change the appliance cord to a three-wire, making the connections inside according to the manufacturer's instructions for a three-wire cord.  It's a simple fix, easy, inexpensive, code compliant, and reasonably safe.  There are literally millions of ranges and dryers installed this way in older homes in the US.
With most ranges and dryers, the instructions will show how to connect your cord if you're using a receptacle without an equipment ground.  The connection generally bonds the neutral to the frame of the appliance.  While this is not quite as safe as a four-wire circuit wired with an equipment ground, it does provide protection / fault clearing in case of an internal ground fault in the appliance.  Here's an example:

from a power company handout.
If you have a four wire cord with the usual four-wire connections inside the appliance, but the equipment ground goes nowhere in that receptacle, you have no fault clearing for those internal ground faults, which is potentially dangerous.
If someone knew what they were doing, I suppose they could use a four-wire cord with a four-wire receptacle that has no equipment ground, but bond neutral to ground inside the appliance like you would with a three-wire cord.  You would have the same fault-clearing you'd have with a three wire cord.  But that's still a bad idea.  For one thing, if anyone ever corrects the wiring of the receptacle, but assumes the appliance is wired normally - a reasonable assumption - they will not remove the bond / jumper inside, and it will create a dangerous configuration where you have unwanted current on the equipment ground.  But if they know what they're doing, wouldn't they just do it the right way?
In addition, since the instructions for the four-wire receptacle don't make that equipment ground optional, it's a code violation to install the receptacle without the ground.  Even if it was not a code violation, it's a common sense violation.  When someone sees a NEMA 14-50R it's reasonable for them to assume there's an equipment ground, and if they plug in equipment wired for that equipment ground, they will unknowingly create an unsafe situation.