On the back side of the Teensy 4, there appears to be a pad layout for an SD socket:
https://www.pjrc.com/store/teensy40.htmlThe Teensy 4 might be out of the power budget: It requires 100 mA when running full speed.
Using the Arduino IDE would be my absolute next-to-last choice. The only IDE that is worse is chisel and stone tablet. OTOH, once TeensyDuino is installed, the Arduino IDE will sort out which toolchain to use and that's a handy feature. It uses a particularly regrettable color scheme and, via Google, there are alternatives that don't display the error messages in invisible red-orange against black. It also doesn't support the notion of multiple files in a hierarchy or tree structure. That chisel looks better and better.
There is an add-in tool for Microsoft Visual Studio called Visual Micro that provides an identical Arduino capability with a more professional IDE. I don't know if it works with the TeensyDuino toolchain. There is a difference between Microsoft Visual Studio and Microsoft Visual Studio Code - discussed after Eclipse.
I hesitate to recommend Eclipse simply because start-up configuration can be a challenge. It's a nice IDE but it might be an advanced topic.
I have been using Visual Studio Code for a Lattice FPGA project and it's pretty nice. There is an Arduino add-in but, again, I don't know if it works with TeensyDuino but it does allow for other ARM32 chips so I don't see why it wouldn't. It requires the complete Arduino IDE and TeensyDuino is an add-on to that environment so it probably shows up in Visual Studio Code. Do a test: Add TeensyDuino to your Arduino installation, install Visual Studio Code (Community Edition) and then install the Arduino extension. See how it all works out. VS Code is a very advanced editor - way beyond my need, but it's pretty nice.
The only problem I have with VS Code is that it doesn't know how to print - anything. One of the gurus pointed out that he had never needed to print his code for mark-up or review. Well, goody for him! I'm a mere mortal, I need printed copies. Not a big deal, I can use Notepad++ to open the files and print them. Actually, Notepad++ is a pretty nice editor and, while I haven't chased it down a rabbit hole, it appears it can be used with Arduino. Google has some info.
Lots of choices and the Arduino IDE isn't in my selection set.