In BJJ and other martial arts, I've seen people with black belts that have been customized, like having words written on them, etc. I was wondering if it is okay for someone who has a color belt to get a custom one? For instance, if someone just got their purple belt, would it be okay for them to order a new one before they get any stripes?
I would be very wary with this: Some styles, rather than having a stripe across the belt at one end, have a stripe along the entire length of the belt. If you get your belt "customised", you may be seen as trying to pretend a higher grade than you actually have. Even if not, it may be regarded as egotistical or disrespectful.
More importantly: Where I live, there are a number of different martial arts clubs. Different clubs and styles get together occasionally for training sessions on "common-ground" - stuff like free-sparring, or street-based self defence (recognising potential situations, staying within bounds of law, et cetera) - which is style-agnostic, or benefits from exposure to an unfamiliar style.
On thing common to all of these clubs is: People very rarely buy their own decorated belts. They are (with one, notable, exception) a gift from the club or instructors, as a sign of respect or recognition of skill and accomplishment. Self-bought belts, even black, tend to be unadorned aside from stripes.
As said, there is one club who are an exception to this rule. Their Dan grades (black belts) are not distinguished by a gold bar / stripe on the end of the belt until 4th-Dan. For the first 3 degrees of black belt, they have the Club and Style name embroidered on their belt in Blue, Red, or Green. This is their equivalent to a plain, undecorated black belt with stripes.
At 4th-Dan (instructor grade, requires coaching certificate), they are presented with a belt which has the Club name, their name, and 4 bars, all embroidered in gold thread.
For this question there is only one person to ask: Your trainer(s).
Is it okay for them or not?
If it is okay for them, always have a normal belt, just in case you do classes or training somewhere else.
Have a look at the class. Do they all wear traditional white gis and plain belts, some kids maybe striped belts? Then the answer will be rather negative. Or something like "If you need it urgendly, you may do it."
Do they wear anything and it doesn't look traditional, then better still ask. The answer will be rather: "It's ok."