A SIM card reader gets information from the SIM card. Nothing more and nothing less. Just having that information does not mean you can use "mobile data".
The question becomes what do you intend to do with that information? How do you intend to exchange that information with someone who can allow you to use mobile data?
The SIM card does not contain a mobile phone radio, it just contains the information used by a mobile phone radio to connect to a mobile phone network.
The frequencies used by mobile networks are very different from those used by Wifi or Bluetooth, and the encoding of data and power levels used are also very different. None of the "normal" radios present in a PC can communicate with cellphone towers as they lack the radios and protocols necessary.
A SIM Card Reader is only a small part of the hardware necessary to allow a computer to talk to cellphone towers and so "use mobile data"
You can get LTE adapters (or laptops that already contain LTE modules) that contain all the radio electronics necessary to talk to a cellphone tower, but normally these also contain a SIM card slot so your SIM Card Reader is completely redundant and useless in this case.
So no, a SIM card reader alone is not enough.