What you are looking at is a rubber dome switch. These are typically custom parts, and the minimum order quantity (and design costs) are likely to be out of your price range.
Depending on your requirements, you have a couple of options:
Some rubber dome switches are available in standard PCB-mountable packages (containing both the rubber dome and the switch contacts). The sizes and shapes will be limited, though.
Reuse parts from an existing rubber dome switch. For instance, if you needed one just like the one in the picture, you could cannibalize the switches from the original controller. If you needed to position the switches differently, you could cut apart the rubber between the domes and glue the resulting parts in place.
Consider using a tactile dome switch instead. These switches use a buckling metal dome instead of a rubber part. The metal dome is a reasonably standard part, and can be positioned on the board with adhesive tape. One caveat is that these switches are much "clickier" than rubber domes.
If you just need to detect a finger touch without pressure, you could use capacitive touch sensing. This will require a conductive surface on the switch.
I have seen (in fact I bought a bag of 1,000 of them) individual (none-bespoke) rubber switches that include conductive rubber and dome for a single switch.
Like most other electronics bits, they should be easily available in China. Photo from a random Alibaba supplier.
It appears to be a moulded conductive rubber button. The two black strips are conductive rubber that bridge contacts on the PCB it fits over. These are typically custom devices, although Brady or someone might make some simple strip and matrix devices.