I'm new to the game and relatively new to speedrunning altogether, so bear with me on this. When I'm trying to execute a dash, sometimes it just won't come out when enemies are close to me. I made sure it's set to manual in settings, so I know it's not that. It only seems to work 100 percent when there's nobody within close range. Is this a mechanic of the game or am I just doing it wrong?

a lot of times if you try to start a dash maneuver when too close to an enemy you will get hit or grabbed, particularly if you are flipping to set up a slide or tackle. The timing of the double tap can be a little tricky too, especially if you try to start an attack or slide. Too early on the attack and it will override the dash.

Welcome to the community! Definitely drop by the discord, there is a ton of knowledge everyone is happy to share!

I have also found that you are playing on a controller and touch up or down at all, it will cancel the dash. Just a thought!

The dashing in TMNTIV is a bit finicky and there's a lot to get used to. Being close to enemies doesn't affect the dashing but you're probably moving around a lot when they are close.

When you tap forward, whether it's a really fast tap or you actually move a little bit, on the moment of input the game waits for about 1-1.5 sec to see if you follow it up with another forward input in order to execute a dash.

This means that if you walk forward a bit and then double-tap within that 1-1.5 sec you will most likely cancel out the dash with one input too many, which results in giving a regular walk.
If you walk a tiny bit and then try to run you want to just tap forward once.

This also creates some inconsistencies as far as dashing moves are concerned. If you do F > F > SP for a dash special, the dash special animation is so long that you lose the dashing state when the attack is over, which means you will need to do F > F > SP again in order to do another dash special.
But with slides and tackles you will only need one forward input after doing the first one because those are fast and you're still in dashing state after them. So for example first slide would be F > F > JMP > ATK, and then you do, F > JMP > ATK, F > JMP > ATK, etc.
Dashing and then turning around to dash is a pain too as you gotta triple tap for some reason.

As already stated by matthewreality, luckily you can cancel this small period by moving up or down, or by doing a regular attack. So if you want to make sure you get the dash just do any of those.
If you are close to enemies an attack is recommended for the short stun.

I'd recommend just dashing along the edge of the screen for a while while there are no enemies around to get a hang for it. As your strats become more solid and less improvised the dashing mechanics will feel way better.