Let's say my opponent controls a Spirit of the Hearth, and thus has hexproof.
I summon a Burning Sun's Avatar, and its enters-the-battlefield ability triggers.
I don't have a legal target for target opponent, but I could for up to one target creature.
Does the triggered ability make it to the stack with as many legal targets as possible, or does it get cancelled by not being able to choose a legal target when putting it on 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.
No Burning Sun's Avatar's enter the battlefield ability will not resolve, so no damage will be dealt to the creatures.
In order for an ability to be put on the stack it requires legal targets. Burning Sun's Avatar ETB specifies (emphasis mine):
When Burning Sun's Avatar enters the battlefield, it deals 3 damage to target opponent and 3 damage to up to one target creature.
If you can not target one opponent and either one or zero creatures the ability will not go on the stack. This is covered in the comprehensive rules under here (putting the ability on the stack is the same as casting a spell, emphasis mine):
601.2.: To cast a spell is to take it from where it is (usually the hand), put it on the stack, and pay its costs, so that it will eventually resolve and have its effect. Casting a spell includes proposal of the spell (rules 601.2a–d) and determination and payment of costs (rules 601.2f–h). To cast a spell, a player follows the steps listed below, in order. A player must be legally allowed to cast the spell to begin this process (see rule 601.3), ignoring any effect that would prohibit that spell from being cast based on information determined during that spell’s proposal. (Such effects are considered during the check detailed in rule 601.2e.) If, at any point during the casting of a spell, a player is unable to comply with any of the steps listed below, the casting of the spell is illegal; the game returns to the moment before the casting of that spell was proposed (see rule 720, “Handling Illegal Actions”).
and
601.2c: The player announces his or her choice of an appropriate player, object, or zone for each target the spell requires. A spell may require some targets only if an alternative or additional cost (such as a buyback or kicker cost), or a particular mode, was chosen for it; otherwise, the spell is cast as though it did not require those targets. If the spell has a variable number of targets, the player announces how many targets he or she will choose before he or she announces those targets. In some cases, the number of targets will be defined by the spell’s text. Once the number of targets the spell has is determined, that number doesn’t change, even if the information used to determine the number of targets does. The same target can’t be chosen multiple times for any one instance of the word “target” on the spell. However, if the spell uses the word “target” in multiple places, the same object, player, or zone can be chosen once for each instance of the word “target” (as long as it fits the targeting criteria). If any effects say that an object or player must be chosen as a target, the player chooses targets so that he or she obeys the maximum possible number of such effects without violating any rules or effects that say that an object or player can’t be chosen as a target. The chosen players, objects, and/or zones each become a target of that spell. (Any abilities that trigger when those players, objects, and/or zones become the target of a spell trigger at this point; they’ll wait to be put on the stack until the spell has finished being cast.)
If the ability of Burning Sun’s Avatar has two targets and one becomes illegal, the other will still be dealt damage.
