There are tons of Free QR code creators (or low cost).
You could create a coupon, assign unique QR codes for each affiliate (place them on cars), and scan them in store to give proper credit to the affiliates.

The only problem I see with this is: human nature.
We forget things, lose things, and show up asking about "that deal that some guy told me about."

Or

Personally, I would want email addresses to communicate with the folks. (to really measure the effectiveness of this promotion)
I'd do this:

-Create a different bitly (www.bitly.com) link for each affiliate.
-Place them on a coupon that has a great incentive.
-Write on there- redeem your coupon by visiting (the bitly link)
-This should take them to a simple opt-in page (to gather email addresses- you can use www.leadpages.net)
-They give you their email, you deliver their coupon AND you are able to continue communicating. (since most sales take 7 touches before a decision is made anyway)
-AND your affiliates get the credit without error.

You'll get the most qualified leads and you can judge for yourself how effective this will be moving forward. :)

This might be a completely different approach however this could help you in the long run.
Develop a simple mobile application for your vendors who would refer customers to you .

1. Customer visits the kiosk , sales person refers and inputs customer info
2. The application sends out digital business card to customers
3. When customers need your service their contact info is already in your system and the referrer gets commission.
4. Every time a vendor enters customer info on the application it updates the database that you can access through a admin portal.

In a way you can also use customer info to send out any future coupons , ads etc to potential customers.

Go with a very basic application so your investment is low and later improve the application based on the ROI

There are other approaches to this as well however if you are up for a little investment this may turn out to be helpful.

Have you thought about handing out coupons to these partners with unique Coupon Codes for each partner? While your barcode idea is similar, the coupon could motivate prospective customers to value the referral even more.