How to check if all parts of a shape file were uploaded to FME server?

You could use the FME Server REST API to check what files are in a certain location.

The structure could be:

https://<your server>/fmerest/v3/resources/connections/FME_SHAREDRESOURCE_DATA/<files or folders>

With that response, you can check that all files belonging to the shapefile are present by using an Aggregator and a TestFilter.

For example: 'test.shp', 'test.dbf' and 'test.shx'. Extract the filename (test) and the extension (.shp) with a stringsearcher and regular expressions. Then use an Aggregator based on the filename and concatenate all the extensions in one attribute. Then use a TestFilter to check if the extensions (.shp, .dbf, ...) are in this concatenated string.

Another solution would be to start working with SHZ-files, i.e. a ZIP-file of the necessary files of a shapefile, with the extension changed to SHZ in stead of ZIP. This way only a single file needs to be uploaded.

Shapefile is a nasty format in that respect, you need the .shp, .shx and .dbf and the rest is optional. I agree with @lars_de_vries that a .shz (or a .zip) is the easiest way to solve this. At least it'll solve the "is it uploaded or not" problem. You don't need to change anything to your workspace even, just make sure you let your users know they should upload a zipfile. However, it's probably wise to add a message when the workspace fails to explain what the workspace expects in terms of input.

Another solution would be to start working with SHZ-files, i.e. a ZIP-file of the necessary files of a shapefile, with the extension changed to SHZ in stead of ZIP. This way only a single file needs to be uploaded.