You will be able to access GCP services as long as you have a browser, or an RDP client for accessing Windows instances on GCP, or an SSH client for accessing Linux instances on GCP.
You won't be able to install the Cloud SDK locally, but the gcloud, gsutil, and bq command-line tools are available to you in your browser through the GCP Console.
SSH Client Options: MobaXterm has a portable client that will let you run SSH in a Linux like command line from Windows. If you can run their EXE file without admin rights on your PC, you are good to go. I believe Putty also has a portable client.