Verbjorns Ljosa's answer only includes permissions for cloudwatch to invoke the lambda. Have you specified the proper policy and iam role that allows the lambda to perform its actions?
resource "aws_iam_role" "check_foo_role" {
  name="check-foo-assume-role"
  assume_role_policy="assume_role_policy.json"
}
with assume_role_policy.json
{
  "Version": "2012-10-17",
  "Statement": [
    {
      "Action": "sts:AssumeRole",
      "Principal": {
        "Service": "lambda.amazonaws.com"
      },
      "Effect": "Allow",
      "Sid": ""
    }
  ]
}
and a policy referencing the above resource iam role I.e. something like
resource "iam_role_policy" "check-foo-policy" {
  name="check-foo-lambda-policy"
  # referencing the iam role above
  role="${aws_iam_role.check_foo_role.id}"
  policy="check-foo-policy.json"
}
and finally the json specifying  the policy, check-foo-policy.json.
{
  "Version": "2012-10-17",
  "Statement": [
    {
  "Effect": "Allow",
  "Action": [
    "logs:CreateLogGroup",
    "logs:CreateLogStream",
    "logs:PutLogEvents"
  ],
  "Resource": ["*"]
},
{
  "Effect": "Allow",
  "Action": [
    "abc:SomeAction",
    "abc:AnotherAction",
  ],
  "Resource": "some-arn-matching-the-actions"
}
Do note that you cannot specify a Resource restriction for the logs-related actions. abc:SomeAction might be ssm:GetParameter with an accompanying resource arn like "arn:aws:ssm:us-east-1:${your-aws-account-id}:parameter/some/parameter/path/*