With the GitHub API, how do I find out the sha1 for the file tree as it existed at a given version/tag?

The GitHub API documentation explains how to retrieve a repo's tree of files using the endpoint

/repos/{owner}/{repo}/git/trees/{tree_sha}

Given the version tag of a release (say, v1.0), how do I figure out the corresponding tree_sha value so that I can use the API endpoint to get the tree of files at that version of the repo?

You can look up a tag (like "V1.0") using the "References" API like this (I'm using the gh cli tool, but you can access the same APIs using your language of choice):

$ gh api /repos/cli/cli/git/ref/tags/v2.27.0
{
  "ref": "refs/tags/v2.27.0",
  "node_id": "MDM6UmVmMjEyNjEzMDQ5OnJlZnMvdGFncy92Mi4yNy4w",
  "url": "https://api.github.com/repos/cli/cli/git/refs/tags/v2.27.0",
  "object": {
    "sha": "89caedf1817c21e4bdb9b136a46444225ec4c982",
    "type": "commit",
    "url": "https://api.github.com/repos/cli/cli/git/commits/89caedf1817c21e4bdb9b136a46444225ec4c982"
  }
}

That gets you the commit sha. You can get information about the commit using the commits API, which will give you the tree sha for that commit:

$ gh api /repos/cli/cli/git/commits/89caedf1817c21e4bdb9b136a46444225ec4c982
{
  [...]
  "tree": {
    "sha": "3d16aa8873d6b1201705fa450f14f2f43c61412f",
    "url": "https://api.github.com/repos/cli/cli/git/trees/3d16aa8873d6b1201705fa450f14f2f43c61412f"
  },
  [...]
}

And now you can ask for that tree:

$ gh api /repos/cli/cli/git/trees/3d16aa8873d6b1201705fa450f14f2f43c61412f
{
  "sha": "3d16aa8873d6b1201705fa450f14f2f43c61412f",
  "url": "https://api.github.com/repos/cli/cli/git/trees/3d16aa8873d6b1201705fa450f14f2f43c61412f",
  "tree": [
    {
      "path": ".devcontainer",
      "mode": "040000",
      "type": "tree",
      "sha": "aa03b789b096f7e044451a9069aa11abdd6e3882",
      "url": "https://api.github.com/repos/cli/cli/git/trees/aa03b789b096f7e044451a9069aa11abdd6e3882"
    },
    [...]
  ],
  "truncated": false
}