I wanna get into SMG2 Speedrunning but I just want to know if Dolphin Emulator is allowed for Any% runs, or am I going to need a capture card?

it does seem to be allowed, just look at the rules.

From the rules:

Emulation Restrictions/Requirements:

• Only Dolphin emulator is allowed.
• Only beta versions and Lua Dolphin are allowed.
• You must have FPS shown either in windowed mode or on the game itself in fullscreen mode. If you choose to play in fullscreen mode, you must show the Dolphin version at some point in the video. This can be shown after the run for convenience.
• The "speed up disc transfer rate" option must either be unchecked (blank) or be set to undetermined (-) and must be shown at some point in the video. All runs except individual level runs must show this option. This can be shown after the run for convenience.
• The game must be running at a smooth 60 fps. Occasional frame drop/lag spikes are acceptable however, we may reject runs that consistently run at a lower framerate or drop frames frequently.
• Slowing down the game in any form is banned.
• Speeding up loads in any form is banned.
• Save states are banned, however they may be used outside of runs, i.e. resetting runs.
• You may use any form of controller.

Small note (might gonna suggest to mod to change it) that "speed up disc transfer rate" name has changed to "Emulate Disc speed" name a few a months ago in case you didnt find it