How do you find records on someone who disappeared in the early 1900's?

You could check the various genealogy online search sites, newspaper searches, obituary searches, military records- many of which have trial memberships you could use for the purpose of checking that particular person. You could check through the United States census records and if you find the party that will give their last known address and other persons they may have lived with. Then, you could do a more focused search in that particular geographical area- death records, newspapers, marriage records, school records, etc. would be a good resource to explore.

It depends on what you mean by 'disappeared?'

I'd check census records, death records, the SSDI.

If I had more details, I'd have a better idea of what records would be the most promising.

if you know where he/she was last seen then you could serch that areas death and medical records to find a bit of info.