After hours of searching for a solution to a similar problem, I found this page in the documentation about the script step of Action Designer. Hidden half-way down the page is the following line.
The Script step always converts data stored in the inputs and outputs global objects into strings.
So actions can have objects as inputs, but the script step flattens them before bringing them into the script. This means that before you can call record.variables, you need to convert record from a string into a GlideRecord. For anyone reading this in the future, this can be done by adding the following lines before calling record.variables.
var record = new GlideRecord('sc_req_item'); //consider passing this as an input instead of hard coding
record.get(inputs.record); //inputs.record is now a sys_id. We are converting it to a request record.
record.variables[inputs.variable] = inputs.value;
If working with JSON objects, like I was, call JSON.parse(input.record) to rebuild as an object.
