I have requested countless times vegan/vegetarian meals on long-haul flights.
Normally they are pretty well organized and you won’t have to ask for the meal. My strategy is just to wait and if given the normal food, then I politely remind them I have a special arrangement. In hundred of flights, this has only happened maybe 1 or 2 times.
In most airlines, they deliver the special meals ahead of time and only then will they start delivering the normal menus. It is a rare ocurrence that the crew will make the effort of delivering your meal in sync with everyone else [Turkish Airlines however managed delivering in sync pretty well].
I will also add that often in the vegetarian menu, the desserts and breakfasts are far nicer than the alternatives, as they won’t give you that terrible egg substitute and you often get real fruit instead of the boring generic cakes. [and I prefer to know what I am eating rather than eating something I do not know how it was made ]
The experience may vary because often the service crew/crew does not understand the concept and nuances of vegetarianism, from ovo-lacto to strict vegetarianism. From my experience, in the UK and middle east airlines often they got it right as they frequently have Indian/Hindu customers.
Also from the voice of experience, the request has to be done at booking time. By check-in time, the logistics are already in motion and they won’t be able to cater to your preferences.
As @happybuddha correctly points out, it is always sensible to ask at check in/boarding time wether you have the meal in place or not. However, if there was a fault it will not always be corrected. The process is unreliable at best, depends on the willingness of the parties involved, and of a member of the crew taking the time to coordinate/fetch your meal from the service crew and carrying it by hand to the plane.
You may however have the luck of them having a extra veg meal by some other motive. Once I asked why I was not given my veg meal, and whilst I was told it was not in the system, I was asked if I would not mind having a hindu meal instead.
Be aware, when a plane is “upgraded” at the last minute, or there is a malfunction and there is a plane change, all bets are off. There is a strong possibility that they won’t have time to deal with the logistics and will just load a generic meal for x passengers.
If you miss a flight/connection, forget about your special meal. As I said before, at check-in time they won’t be able to change meals, even though they may say they will to try to keep you happy – in a small airport/company someone of the crew might be tasked to carry the meal, it already happened with me, but it was a courtesy and it is not a common occurrence. The meals are usually already in the plane, or already prepared to be loaded in the plane by that time. It seems pretty much a requirement in many airlines that any special meal has to be booked at least 48h in advance.
Even for non-vegetarians it may be worth looking at special meals – for instance Emirates Airlines and Turkish airlines have a wide choice, including several ethnic choices and sea food.
Please do note that in some short-haul flights, you might have hot meals as well if you buy the meal at booking time, like in Cebu Pacific or Cathay Pacific airlines. The price can be competitive compared to having a meal/snack at a restaurant/coffee shop in the airport, and it also can save you valuable time.