How to maintain a receiving and production system for recall?

We are a small dumpling production company. We are trying to keep records of all raw materials received and used during the production process, however, a lot of raw materials that we use such as cabbage, onions, cilantro and tomatoes do not have lot codes/ batch numbers and they also don't have best by date or an expiration date. Could someone suggest some ideas to keep track of the raw materials that do not have the lot codes or an expiration date? We are having difficulty with creating a system to keep track of raw materials as receiving is done by one individual while the preparation of the raw materials for the production is done by another and the production is done by another individual. We also have to separate production for veggie and chicken dumplings. Chicken and veggie dumplings are made alternately. And there are different people for preparing the ingredients for the production (for example: there is one person to cut onion and someone else preps the chicken and then someone else preps the cabbage, there is a different person who mixes the ingredients and production is done by someone else). In this case, how should I go about creating a system of keeping track of raw materials as the person who does the production won't be able to record the lot code as they just use the already prepared raw materials.

You could / can assign the product a lot number at receipt.   It could include a bill of lading number, po, number, sales order number, etc  Or it could be completely arbitrary.  as long as you know that that number can be tracked to a specific shipment / receipt.  Then tag the product with that lot number.  

After the ingredient is "prepped" you could assign it a new lot number.  again this could have some meaning such as Julian date prepped , etc., Or it could be arbitrary.  The prepped ingredient is then tagged with the new lot number.  You wouldn't necessarily need to assign a new number and could use the same lot number as the raw lot number.  

when the final product is assembled, the lot number of the raw material or prepped product is recorded and the final product is given a lot number.  The final product is the given a new lot number.  

these are just suggestions as its hard to understand the flow of production at your facility.  the jest is that you have a way to track each of the ingredient used back to a shipment.