AWS
Because meeting the specific needs of the build environment would require a dedicated host on AWS the cost becomes prohibitive almost immediately. The least expensive option (the compute-optimized c4 dedicated host) alone would cost between 37k and 72k over six years before paying for storage or bandwidth.
Azure
Azure is in a similar situation. Their compute-focused VM instances start at about $100 a month each meaning that even keeping one active over six years would top $3.7k. Running 8 VMs and a single SQL Server for that period tops $32k before even paying for persistent storage or bandwidth.
Local
Either option proposed for a local alternative is considerably less expensive than the cloud-based options and would represent gigantic performance upgrade over the existing solution. In addition the process of transitioning would be considerably less complex.
At this point I recommend keeping the build environment in house at this time.
Which is sad in the sense that I was excited about the prospect of a cloud deploy but that's just not really feasible with the requirements of this particular location.
It's good, though, in that I'm still on track to do a migration and that's very exciting. I've requested a new server with:
A 10-core hyper-threaded processor
32 GB RAM
2x300 GB SAS Drives for an OS (RAID 1)
2x2 TB SAS Drives for VM storage and file sharing
Whether I get all that or not is yet to be determined but I'm excited to get started on the next phase of the project.