I have some experience in this area.
Generally Teensy are socketed rather than soldered directly to the PCB for easier serviceability. Have JLCPCB install sockets for you and install the Teensy yourself since it just needs to be plugged in.
As already mentioned, you can modify the Teensy yourself or it can be ordered in whatever configuration you want from ProtoSupplies.com. If you are making product for sale you will want to ensure the quality is good to avoid problems in the field. If you do the rework yourself, I suggest making a bit of a fixture to hold good connector alignment and also have a good magnification setup so you can be sure of the quality of the soldering and not rely on testing to try to find poor soldering joints.
Be aware that the regular IC style sockets like shown above will not work with the large 0.064" square Teensy pins. You need to use a couple of 1x24 female header sockets like shown below. JLCPCB carries these, but they are not always in stock.
For connecting Ethernet, as already mentioned a cable can be used with the male header on top of the Teensy and connect it to a similar header on your baseboard as below. This works fine, but the connection is not a locking style, so the cable could come loose under vibration or shock
A better solution is to use a female header socket on the bottom of the Teensy which mates with an extended reach male header that is on your baseboard. The extended reach male header is required to make the stack-up work with the female header sockets shown above. The problem with this approach is that JLCPCB won't have these special length male headers. They are available from PJRC or ProtoSupplies.com but difficult to find anywhere else.
As an alternative, you can swap the male/female headers, so the Teensy looks like shown below. In that case, JLCPCB has a compatible female mating connector they can solder for you. This version of Teensy with the extended reach male header is currently made for
AgOpenGPS applications (robotic harvesting equipment).
If you need a lower profile setup, then shorter pins can be installed on the Teensy to mate with shorter female header sockets. In that case, the stack-up works out so that a regular length male Ethernet header can be used rather than the special length one. I don't think JLCPCB has short mating female headers, so if you went this route, you would probably need to install the female headers yourself. I wouldn't go this direction unless you really need the shortest height possible.
--- Ken
ProtoSupplies.com