Honestly, I think your best value-bet for the time being would be to either use your iPhone as a sound recorder on voice memo mode, or possibly just shoot your whole video on it (for picture and sound) - it looks like the Coolpix has a fairly narrow field of view on video mode compared to an iPhone; a wider lens may be better for action anyway?
If you’re using two devices at one time (say, the Coolpix for video and iPhone for audio), just do a clap slate at the start of each take - have someone on-camera say what take it is, then clap where the camera can clearly see it - you can use that to sync the picture and sound later (visually, you have a clap you can see, and in your audio file, you should have a loud, clean waveform of a clap - match the two up and you’re good).
What you also may want to consider is doing a foley pass after the video is shot, and not worry about the sound while you shoot - while this will add some editorial work, it could potentially speed up your actual shoot a lot by reducing the number of complications. If you go this route, you can use the iPhone’s voice memo mode to record some “wild” foley either between takes, or after the video is edited - just put the phone in a good recording position (make sure to shelter it from wind that’ll audibly “buffet” or blow against the mic) and re-create the action you need sound for, maybe in a couple takes. If you have music throughout, this may be sufficient to just cover the key moments that feel big and/or happen in close-up, where you really want to hear stuff clearly - otherwise, whatever sound you capture raw at a low level under the music should work okay for the rest, in most cases. This approach always necessitates some extra editing to get it to sync to picture, but if you’re comfortable with some detailed editing, it can be done relatively quickly.
The other option, which I don’t actually recommend (but am mentioning for the sake of a more complete answer) is buying a low-end handheld recorder like the Zoom H1, which goes for $99.99, exactly at your budget max. Something like the H1 is lightweight and easy-to-use, with built-in mics for recording, but long-term doesn’t allow any flexibility for upgrading, and it also can’t be used as a recording unit for most professional microphones, since it lacks XLR inputs (XLR being the cable type basically all pro mics run on). So while it has some benefits over the iPhone as a recorder, for the time being I’d suggest you save the money for further down the road, unless you try the iPhone and it legitimately cannot do what you need it to do, period.