Can I pass a string variable to jq not the file?

Use the bash: echo "$json_data" | jq '.key'

The value of the variable "json_data" that was given in the original question was not valid JSON, so this response still covers both cases (nearly-valid and valid JSON).

Valid JSON

If "$json_data" does hold a valid JSON value, then here are two alternatives not mentioned elsewhere on this page.

--argjson

For example:

 jq -n --argjson data "$json_data" '$data.key'

env

If the shell variable is not aleady an environment variable:

json_data="$json_data" jq -n 'env.json_data | fromjson.key'

Nearly-valid JSON

If indeed $json_data is invalid as JSON but valid as a jq expression, then you could adopt the tactic illustrated by the following transcript:

$ json_data='{key:"value"}'
$ jq -n "$json_data" | jq .key
"value"

I would suggest using a bash here string. e.g.

jq '.key' <<< "$json_data"