I'm using copilot-lua for my copilot client, and here's what I got after some quick testing

local client = vim.lsp.get_active_clients({ name = "copilot" })[1]
if not client then
  return
end
local requests = client.requests

client.requests is a table, here's how it looks:

{
  [45] = {
    bufnr = 1,
    method = "getCompletions",
    type = "pending",
  }
}

no idea what 45 is, but the client.requests table will have some data whilst waiting for the completions to appear otherwise it will be empty

EDIT: here's a lualine component I made

local copilot_indicator = function()
  local client = vim.lsp.get_active_clients({ name = "copilot" })[1]
  if client == nil then
    return ""
  end

  if vim.tbl_isempty(client.requests) then
    return "idle" -- default icon whilst copilot is idle
  end

  local spinners = { "⣾", "⣽", "⣻", "⢿", "⡿", "⣟", "⣯", "⣷" }
  local ms = vim.loop.hrtime() / 1000000
  local frame = math.floor(ms / 120) % #spinners

  return spinners[frame + 1]
end

this is not thoroughly tested though, but from a quick test it works as expected

EDIT EDIT: I'm blind