Does a triggered ability with no legal target for only some of its targets make it to the stack?
You cannot choose targets for Burning Sun's Avatar's triggered ability, and therefore cannot put it on the stack, if you have no opponents that are legal targets.
Burning Sun's Avatar's ability says

When Burning Sun's Avatar enters the battlefield, it deals 3 damage to target opponent and 3 damage to up to one target creature.

This means that in the process of putting the ability on the stack, you must choose exactly one opponent as the the first target, and zero or one creatures as the second target. If no opponents are legal targets, then you cannot choose a legal combination of targets for the ability, and it will not be put on the stack.
We have the following rules that describe how to put a triggered ability on the stack:

603.3d The remainder of the process for putting a triggered ability on the stack is identical to the process for casting a spell listed in rules 601.2c–d. If a choice is required when the triggered ability goes on the stack but no legal choices can be made for it, or if a rule or a continuous effect otherwise makes the ability illegal, the ability is simply removed from the stack.
601.2c The player announces his or her choice of an appropriate player, object, or zone for each target the spell requires. [...]

If for one of the targets the spell requires there are no legal choices, then the ability is simply removed from the stack.