Can't you do this with a query? Make your own expressions Value1: Part*Weight , Value2:.. Then Add up the values like SumValue: Value1 + Value2... then do the division WeightedAvg: SumValue/SumWeight

How do you figure the WA Price now?  What criteria are you using?
If you can write out a formula on how you calculate the price manually it would be easier for us to help you convert that to something you can use in Access.

The result I am hoping for is also the calculation following:
Purchase Date                                   Quantity             Cost       Extended Cost      Weighted Average*moving
1/1/22 (as an example for layer 1):         10                    2            20.00
2/1/22                                                 20                    2            40.00               (sum of ext cost/sum of quantity) $2.00
3/1/22                                                 20                     3           60.00                       $2.40
4/1/22                                                  30                    4          120.00                      $3.00

As you can see, with each new "layer" created with each new purchase, the weighted average changes. I would like the result to be in columnar form to make the sum easier; all suggestions are welcome! Thank you for any insight you can provide!

So if you are using a query you could make 3 new expressions

SQty: DSUM("Quantity","Table","PurchaseDate<=#" & PurchaseDate & "#")

SCost: DSUM("ExtendedCost","Table","PurchaseDate<=#" & PurchaseDate & "#")

WAvg: SCost/SQty

That should give you the running weighted average as it goes down to current.

You'd also have to add the part to the where condition if it was multiple parts, and you may have to add in the ID field as well if you have multiple same day purchases of that part.

I haven't had a chance to look at this real close but Richard made a video about 3 month ago that I think would help you out.
Check out Moving Average

What are the rules for how your weight changes?