For trace recording and decoding, the MicroTrace must be able to read the program code from your target's memory (which is normally possible) and must have access to your physical trace pins (TRACECLK, TRACEDATA[0..3]) usually via a MIPI20T debug connector.
If both is possible, you can trace your program code. You do not need to have an ELF or any other file containing debug symbols. However, finding any issue is of course a lot more easier, if your have debug symbols (e.g. from an ELF).
I hope your are not using the tiny Arm Mbed LPC1768 Board, because that board does give you direct access to the debug interface (JTAG, SWO) nor to the trace clock, which makes it not suitable for any external debug and trace tool (like MicroTrace).