2003 chevy s10 starter turns but does not fire sometimes

what transmission, here. automatic or manual stick shift?
S10 not blazer S10

old cars this old (pre digital documents) are hard to find and when found are not complete. or poor .
A/T box?
the PRNDL switch must be good. and cranks in both Park and neutral.
yes the relay clicks too, if not the PRNDL is prime bad part.
there is fuse named CRANK.(1st inline here)
the starter can be hot wired (a test) to prove starter work.
if stick shift TRANNY it is called the clutch switch. to crank,
some books say hood up fire wall, center is relay. IDK


see why we ASK transmission first/
if the relay does not click the fuse crank blown
or PRNDL is dead only on P mode, try N.
if stick shift,, clutch CCP switch bad or just mal-adjusted.
ST below goes to the relay then to starter solenoid
the relay colors are there. Green or Purple.

PRNDL is on side of TANNY part of shift linkage.

https://www.google.com/search?rlz=1C1CHBF_enUS812US812&q=1996+chevy+s10+firing+order&tbm=isch&source=univ&sa=X&ved=2ahUKEwj9sqD0kb3jAhXpmOAKHfA9DnsQsAR6BAgGEAE&biw=1600&bih=789 https://www.justanswer.com/chevy/1d7di-firing-order-chevy-s10-pickup-2.html

I doubt if the engine is seized but with the loss of oil in the transmission it could be a seized bush/ bearing in the torque converter or input to the ransmission

Bad ground wires. Inside they will be at the steering column under the dash. And on on each side behind the side kick panels. Also check the battery to body ground and engine to body ground.

Turn the key to on /run position. Remove gas cap and listen for a humming noise. If nothing is heard then the likelihood is the fuel pump is bad. But before removing the fuel tank, check all the relays and fuses for fuel. If that all checks out as good, then check the entire ignition system (cap, rotor, wires, and coil.

Is it happening on all cylinders? Or just one? Have you removed the injector an inspected it. I have never heard of all going bad at once. It would be a good idea to look for open fuses. The main connector is connected properly, each one has good connection.have you changed something lately?

Its possible that you have a bad starter. I would start there, and go from there. good luck

If you remove the starter, you can access the bolts through the starter drive hole... it isn't easy, but definitely possible.

could be a couple of thing,maybe your battery going bad,your ignition switch could be going bad,or possibly your starter.

first you need to check if your are getting spark, then go to fuel pressure, you should have 60lb of pressure