Will I need a new derailleur when changing to a higher ranged cassette?
There are two aspects to the derailer that come into play here. One is total capacity, which is the total amount of variance in chain slack the derailer can accommodate while keeping the chain taut. Single ring setups such as yours with any kind of "traditional" range cassette (say 12-32 as opposed to whats on some modern 1x bikes, like 10-50) come nowhere close to exceeding the total capacity of even a short cage rear derailer, so you're probably good there.

The other factor is how large of a largest cog your rear derailer can clear without running into it and grinding in that gear. Derailers vary on that spec but many road derailers of the last decades max out at 28, while mountain rear derailers all accommodate at least 32 and go up from there depending on the model. There are various hack-type tricks that can buy you a little more clearance, like running the b-tension screw backwards or putting in a longer one, but many derailers still won't go up to 32 even then. The dropout design can also be a factor. Wolf Tooth Roadlinks and similar products are another hack, and exist to make road derailers do exactly what you're trying to do, but they degrade shifting somewhat and for your needs it would be potentially more cost effective to just buy another derailer at that point.

So in other words, if you want to switch to 12-32 and you have a road derailer now, you should probably buy a mountain derailer. The "optimal" choice would a short cage one that still has the large cog clearance you need, which aren't super common but do exist.

Alternatively you might seriously look at whether 53 is the right tooth count up front for what you're doing. Is 53-14 a gear you use to good effect? Will 53-12 be? Another route you could go is a smaller ring, which has the advantage, presuming your cranks have one of the more common bolt circle diameters, that you could get a modern narrow-wide ring and have better chain retention. You can use a gearing calculator to compare what different combinations would give you.