Oh well, another issue with PortableApps.com spec 
It works if you have the complete suite installed, but it's bloatware for most of Dolphin users...
I'll take a look, but please note that even if I drop the need for DirectX Portable, Dolphin would still require DirectX (the portable launcher will take care of it in this case)
EDIT: from now, use this launcher. There is no need for DirectX Portable, but you'll need an internet connection during the install as the needed DLLs will be downloaded at install time (after that you can put the "DolphinEmulatorPortable" folder wherever you want).
Also, you should use only DSound or OpenAL audio backend in the Portable launcher, XAudio2 requires registering a DLL and this can't be done without administrative privileges, so, XAudio2 backend in the portable version will only work if it was already installed locally in the machine you're using (which doesn't seem to be the case, seen that your school laptop lacks even XINPUT1_3.dll)
It works if you have the complete suite installed, but it's bloatware for most of Dolphin users...I'll take a look, but please note that even if I drop the need for DirectX Portable, Dolphin would still require DirectX (the portable launcher will take care of it in this case)
EDIT: from now, use this launcher. There is no need for DirectX Portable, but you'll need an internet connection during the install as the needed DLLs will be downloaded at install time (after that you can put the "DolphinEmulatorPortable" folder wherever you want).
Also, you should use only DSound or OpenAL audio backend in the Portable launcher, XAudio2 requires registering a DLL and this can't be done without administrative privileges, so, XAudio2 backend in the portable version will only work if it was already installed locally in the machine you're using (which doesn't seem to be the case, seen that your school laptop lacks even XINPUT1_3.dll)
Avell A70 MOB: Core i7-11800H, GeForce RTX 3060, 16 GB DDR4-3200, Windows 11 (Insider Preview)
ASRock Z97M OC Formula: Pentium G3258, GeForce GT 440, 16 GB DDR3-1600, Windows 10 (22H2)
ASRock Z97M OC Formula: Pentium G3258, GeForce GT 440, 16 GB DDR3-1600, Windows 10 (22H2)