From the blog post How to….. Loony Dook, it appears the local businesses allow you to use their toilets to change:
Fancy dress for the main event and people go all out for it so there
is no need to feel embarrassed so please make an effort. You will need
lots of layers for beforehand as there will be lots of waiting round
(you can take these off as the parade starts) and a towel and some
clean, dry clothes for afterwards. There will be opportunity to change
in local businesses toilets.
also
At the registration point there are a few pubs and cafes which get
really busy. These open their toilets up to participants and one café
had a stall selling hot drinks, bacon baps, soup etc to help keep you
warm.
The blog also mentions that in the past there have been no lockers available to store your bags/clothes, and recommends having a friend hold onto your items for you. It also suggests you could leave your belongings on the beach, however this comes with a little risk of losing the items, so don't take valuables.
In 2016 there were no lockers available meaning participants had to
leave one person looking after everything or when they arrived on the
beach leave there stuff there. Only participants can get onto the
beach however there is always the risk of people picking up the wrong
bag at the end so many people get waterproof pouches and put their
phone and money in that. Try and limit your valuables you have on the
day.