How to find the PID of an app without ADB? The ADB method is easy enough, but I may not have a computer at hand. Can I retrieve the PID with a Termux package or an app, or any other methods (including those which require root)?
If you are using Android 7 or above, you need to have access to ADB (either command-line client, or an app which used adb on localhost) or root access to see the PIDs of other processes.
If you are in Terminal emulator app or in Termux app, and you have root access, do this:

su
ps -A -o PID,USER,NAME

This would output text segregated in three columns: first column as PID, second column is the user which owns the process, and the third column showing the name of the process.
Example output (posting only a snippet of it here):

28574 system       net.oneplus.odm
28622 u11_a149     com.oneplus.gallery
28646 u0_a149      com.oneplus.gallery
28701 u11_a149     com.oneplus.gallery:recyclebin
28727 u11_a149     com.oneplus.gallery:contentdetector
28778 root         [kworker/1:2H]
28789 u0_a329      su
28794 root         sh
28872 u11_system   com.oneplus.opbackup

Instead of ps command, can also use pidof as suggested by Yahya Uddin here.
Note: PID and UID are altogether different things. If you are after UID, see How can I find app name by UID?