They act as a go-between in that they ultimately host on AWS/GCP. They host their own infrastructure in that they appear to run Kubernetes and have built out their own deployment and service fabric, so they're just using the underlying machines as dumb compute, they're not, eg, building on RDS.
In March 2021, someone asked a question about carbon emissions of their data centres. They said they hosted on both GCP and AWS, but mentioned they were interested in moving to their own bare metal [1].
In April 2021, I asked a question about egress fees to Google, and they walked back a bit the comment about moving to bare metal [2].
As of March 2022, they're still in AWS/GCP [3].
As of September 2022, workloads for new users deploy into AWS, even in regions that were previously served by GCP [4].
[1]: https://community.render.com/t/does-render-use-green-energy/...
[2]: https://community.render.com/t/is-render-com-hosted-in-googl...
[3]: https://community.render.com/t/are-your-servers-owned-by-you...
[4]: https://community.render.com/t/which-render-regions-map-to-w...