My belief is that yes, tulpas are people of their own (and therefore moral patients). My reasoning is as follows.
If I am a person and have a tulpa and they are not a person of their own, then there must either (a) exist some statement which is a requirement for personhood and which is true about me but not true about the tulpa, or (b) the tulpa and I must be the same person.
In the case of (a), tulpas have analogues to emotions, desires, beliefs, personality, sense of identity, and they behave intelligently. They seem to have everything that I care about in a person. Your mileage may vary, but I've thought about this subject a lot and have not been able to find anything that tulpas are missing which seems like it might be an actual requirement for personhood. Note that a useful thought experiment when investigating possible requirements for personhood that tulpas don't meet is to imagine a non-tulpa with an analogous disability, and see if you would still consider the non-tulpa with that disability to be a person.
Now, if we grant that the tulpa is a person, we must still show that (b) is wrong, and that they are not the same person as the their headmate. My argument here is also very simple. I simply observe that tulpas have different emotions, desires, beliefs, personality, and sense of identity than their headmate. Since these are basically all the things I actually care about in a person, it doesn't make sense to say that someone who differs in all those ways is the same. In addition, I don't think that sharing a brain is a good reason to say that they are the same person, for a similar reason to why I wouldn't consider myself to be the same person as an AI that was simulating me inside its own processors.
Obviously, as with all arguments about consciousness and morality, these arguments are not airtight, but I think they show that the personhood of tulpas should not be easily dismissed.
Edit: I've provided my personal definition of the word "tulpa" in my second reply to Slider below. I do not have a precise definition of the word "person", but I challenge readers to try to identify what difference between tulpas and non-tulpas they think would disqualify a tulpa from being a person.
That's fair. I've been trying to keep my statements brief and to the point, and did not consider the audience of people who don't know what tulpas are. Thank you for telling me this.
The word "tulpa" is not precisely defined and there is not necessarily complete agreement about it. However, I have a relatively simple definition which is more precise and more liberal than most definitions (that is, my definition includes everything usually called a tulpa and more, and is not too mysterious), so I'll just use my definition.
It's easiest to first explain my own experience with creating tulpas, then relate my definition to that. Basically, to create tulpas, I think about a personality, beliefs, desires, knowledge, emotions, identity, and a situation. I refer to keeping these things in my mind as forming a "mental model" of a person. Then I let my subconscious figure out what someone like this mental model would do in this situation. Then I update the mental model according to the answer, and repeat the process with the new mental model, in a loop.
In this way I can have conversations with the tulpa, and put them in almost any situation I can imagine.
So I would define a tulpa this way: A tulpa is the combination of information in the brain encoding a mental model of a person, plus the human intelligence computing how the mental model evolves in a human-like way.
My definition is more liberal than most definitions, because most people who agree that tulpas are people seem to make a strong distinction between characters and tulpas, but I don't make a strong distinction and this definition also includes many characters.
And to not really answer your direct questions: I don't know Serial Experiments Lain, and you're the person who's in the best position to figure out if Vax'ildan is a tulpa by my definition. As for "you are your masks", I'm not sure. I know that some people report naturally having multiple personalities and might like the mask metaphor, but I don't personally experience that so I don't have much to say about it, except that it doesn't really fit my experiences.
(I do not create new tulpas anymore for ethical reasons.)