Hello Jason,
We understand that you would like to build and publish Vagrant boxes on a schedule using GCP.
1- Would Google Cloud be a good solution
2- Would I be better off just dedicating a Raspberry Pi or a used laptop PC for this? Depending on the monthly cost of Google Cloud, I'm guessing that I may be better off just dedicating a physical machine of my own for this?
I couldn't find any official GCP documentation about your use case described above. Please contact Vagrant boxes community[1] to help assist you.
3- Can you give me some idea on how much this would cost?
Depends on GCP used resources. Find more details about our pricing visiting the following link [2]. You can also use GCP Pricing Calculator to help you have an estimation[3].
[1] https://www.vagrantup.com/community.html
[2]https://cloud.google.com/pricing/
[3]https://cloud.google.com/products/calculator/