Why there is no "TO" in "having had nobody stand up for me"

I do not understand the following:

Having had nobody stand up for me

Why it is not

Having had nobody to stand up for me

Because:

There is somebody to stand up for me.

The meaning of your statements is actually different:

Having had nobody stand up for me

This means that there was an incident, and no one came to my defense. (Even if they were present in the room.)

Having had nobody to stand up for me

This means there was an incident, but no one was there who could come to my defense. (Either because they weren't nearby, or because I don't have any friends.)

Adding the word to makes a difference in meaning here.

The verb have can take an object with a bare infinitive (without to), just like some similar verbs:

I let her go.

I made her stay inside.

I had her bring me wine.

I had somebody stand up for me.

The last sentence would mean, "I made somebody stand up for me". The above construction (without to) is only possible where have means "make someone do something". However, in your example have does not seem to mean "make", but rather "possess". Of course it's not literally "possess" in the legal sense, but you know what I mean, as in I have someone to clean my house. With "possess", you need to, as you suggested:

I had nobody to stand up for me.

This means "I had nobody whose function it was to defend me".

I had nobody stand up for me.

This means "I made nobody rise from his seat for me". As you see, the two constructions lead to different senses of stand up for being construed.

One thing to note, however, is that the sense of "possess" often also creeps into the sense of "make", so that you can (less formally) also use "possess" without to:

I had my house cleaned. ("made")

I had my car wrecked. (borderline: "I made my car into a wreck", or "I possessed a car that was wrecked")

?This was really cool, I had someone stand up for me at the church when I criticised religion, can you believe it?

Here it means "defend", where the construction is like "make someone do something", but the sense is borderline between that and "I possessed someone who did something". This usage is informal, but it is probably what your example is about. But we can't be sure without context.

If that is how we interpret it, then what is the difference in meaning if you add to? With to, it describes the function of a person: the person to stand up for me is someone who is supposed to stand up for me, or who normally stands up for me. Without to, it just describes a single event where someone did or did not stand up for me.