Some spells can have multiple targets, like eldritch blast (at higher levels) or scorching ray.

Eldritch blast (PHB, pg. 237) says:

The spell creates more than one beam when you reach higher levels: two beams at 5th level, three beams at 11th level, and four beams at 17th level. You can direct the beams at the same target or at different ones. Make a separate attack roll for each beam.

Scorching ray (PHB, pg. 273) says:

You create three rays of fire and hurl them at targets within range. You can hurl them at one target or several.

Make a ranged spell attack for each ray. On a hit, the target takes 2d6 fire damage.

Let's say I ready a spell with multiple targets and that my trigger is something like:

"I attack the orcs (plural) when I can see them"

(Let's assume that my character knows there are orcs hiding behind something and will jump out any moment now; if we need to explain why my PC would know this, perhaps because an ally will do something to draw them out of hiding before the orcs' and my PC's next turn, or perhaps I just keep readying eldritch blast until they finally do come out.)

Given that you can choose not to respond to a trigger for readied actions (PHB, pg. 193):

When the trigger occurs, you can either take your reaction right after the trigger finishes or ignore the trigger.

...then if I ignored the first enemy that appeared but responded to the second, could I target both enemies as I "release" the spell, or would I only be able to target the creature that is the trigger I'm responding to (meaning the second orc in this case)? Or is the trigger not specific enough to be a valid trigger, rendering this whole question moot?

Yes, you can target multiple enemies

The action you take is not dependent on the trigger you set.

First, you decide what perceivable circumstance will trigger your reaction. Then, you chose the action you will take in response to that trigger, or you choose to move up to your speed in response to it.

For example, you could set a trigger of "When an orc comes into view" and set your action to be "heal my ally with this spell".

The only thing you need to make sure is that your action wording does not exclude targeting multiple things. For example a readied action of "target that one orc with eldritch blast" would only allow you to target one orc technically. Whereas a readied action wording of "I hit as many orcs as possible with eldritch blast" would obviously allow you to do target multiple.

Note that you don't even have to specify targets of your spell before you release it. All you have to do is cast the spell with the Ready action and set a trigger for its release. Upon the trigger being met, you can then choose to release the spell and it is at that point that you would choose the target(s) of the spell.

Your case specifically

"I attack the orcs (plural) when I can see them"

Your trigger is being able to see more than one orc. And your action is to attack them. Assuming by "attack" you mean "attack with eldritch blast (which you DM probably understood, but you may want to rephrase it to be clearer), your wording is perfectly acceptable and would allow you to target multiple orcs with the spell if you wanted to. A clearer and technically more correct way to phrase it would be:

I will release eldritch blast when I see one or more orcs.

As long as you are careful with your wording there's no reason you can't target multiple, but it has nothing to do with your trigger condition.

Yes you can

The Ready action states:

First, you decide what perceivable circumstance will trigger your reaction. Then, you choose the action you will take in response to that trigger, or you chose to move up to your speed in response to it.

For the first part you could indeed say "an orc shows itself" and you could pass the trigger until the second orc shows itself. You could also put the trigger on "Two orcs can be hit by my spell" or something similar.

For the second part you can simply state your action is: "I will use Scorching Ray on the triggering orcs/all visible orcs/as many orcs as possible" or if your DM is lenient: simply I will use scoring ray picking the target on use of the spell. Remember here that your action does not need to be targeted at the trigger. You could say "If the hero kicks in the door. I will attack the hostage."