Workaround/Solution

Place a dependency on the aws_api_gateway_integration from the resource causing "NotFoundException: Invalid Integration identifier specified".

resource "aws_api_gateway_integration_response" "click_put" {
  rest_api_id = data.aws_api_gateway_rest_api.mysfit.id
  resource_id = aws_api_gateway_resource.click.id
  http_method = aws_api_gateway_method.click_put.http_method
  status_code = aws_api_gateway_method_response.click_put.status_code
  response_parameters = {
    "method.response.header.Access-Control-Allow-Origin" = "'*'"
  }

  depends_on = [
    aws_api_gateway_integration.click_put
  ]
}

References

There are indications that depends_on aws_api_gateway_integration or placing a wait would be the way.

Probably waiting for the complete completion of the aws_api_gateway_integration resource would be a recommended practice.

• aws_api_gateway_integration_response

  Note: Depends on having aws_api_gateway_integration inside your rest api. To ensure this you might need to add an explicit depends_on for clean runs.

• API gateway integration issue - Invalid Method identifier specified Error #4001

  I'm using the following workaround to solve a similar issue. I'm getting the Invalid Method identifier error on the first run when all the resources are created during the same apply. A proxy API for Lambda is created:

resource "aws_api_gateway_resource" "proxy" {
  rest_api_id = aws_api_gateway_rest_api.rest-api.id
  parent_id   = aws_api_gateway_rest_api.rest-api.root_resource_id
  path_part   = "{proxy+}"
}

resource "null_resource" "method-delay" {
  provisioner "local-exec" {
    command = "sleep 5"
  }
  triggers = {
    response = aws_api_gateway_resource.proxy.id
  }
}

resource "aws_api_gateway_method_response" "response" {
  depends_on = [null_resource.method-delay]
  http_method = "ANY"
  resource_id = aws_api_gateway_resource.proxy.id
  rest_api_id = aws_api_gateway_rest_api.rest-api.id
}

you can just add depends_on. Should be something like:

resource "aws_api_gateway_integration" "create-oauth-lambda" {
  rest_api_id             = aws_api_gateway_rest_api.create-oauth-token-api-gw.id
  resource_id             = aws_api_gateway_resource.auth-token-resource.id
  http_method             = aws_api_gateway_method.method.http_method
  integration_http_method = "POST"
  type                    = "AWS_PROXY"
  uri                     = "some URI"

}

resource "aws_api_gateway_method_response" "response-200" {
  depends_on  = [aws_api_gateway_integration.create-oauth-lambda]
  rest_api_id = aws_api_gateway_rest_api.create-oauth-token-api-gw.id
  resource_id = aws_api_gateway_resource.auth-token-resource.id
  http_method = aws_api_gateway_method.method.http_method
  status_code = "200"
  response_models = {
    "application/json" = "Empty"
  }
}