Or if the pulley is now detached take that with you and you can easily try bolts starting at M6 -- then once you know the size if they don't have grub screws try elsewhere with your new knowledge.
Some grub screws have different end profiles so as you have a sample its easy to find what you need.
See here might or not be helpful
I bought my old CL3 about 35 years ago..... before metric, in the UK - so there's a temptation to assume that we were imperial in those days...... hence the initial thoughts of 3/16 inch.
However, on my machine,
The grub screw needs a 3mm Allen key
There are two screws - but they are not both the same. The lower one ends in a 'top-hat' profile and sits on a shaft key, locking it in place. the Upper screw has a rounded end and locks the lower one in place.
They are both about 7 or 8 mm in length with only a small part threaded - about 3 or 4 mm threaded, however counting these over 1/8 inch against a vernier, comes up with 4 complete turns. Therefore 32 to the inch. It could be 3/16 BSF.
You have a choice of thread here, both at 32 per inch...... depending on the age of the machine.
The CL3 head-stock has not changed much in decades; if it's fairly new then I'd look at a Unified 10-32 as an experiment.
This manual explains the head-stock in some detail- it was used across a range of machines and the screws are mentioned as item 34 in the parts break-down. There are, though some pictures of the grub screws.