The PCs are already actively looking for danger when traveling
You say,
The path of our adventures just so happens to cross this ambush. If they take no active action (from what I've seen around), you make the goblins roll Stealth against the passive perception of the party.
The problem with what you've 'seen around', is that most people simply don't understand how the 5e rules work for active vs. Passive Perception checks, and thus get it wrong. The wording "active" and "passive" is confusing1, so unless you look closely at what the rules actually say, you might also come to the mistaken conclusion that active perception involves a PC 'trying' to look for threats and Passive Perception is just some innate danger sense they have without intentionally focusing on threats. This is just wrong.
Instead, the rules assume that PCs are always actively looking for threats (unless what they are doing is so demanding of their attention that they cannot2) but that they are doing so with Passive Perception checks. In your case, the PCs are moving overland and are already alert for potential ambushes, so if a player says they suspect an ambush and want to 'actively search' for enemies, your answer should be, "You already are".
The rules for this are in the PHB Chapter 8 (Adventuring), under Movement (Noticing Threats). As a DM, if the PCs are traveling, you are expected to compare their Passive Perception scores against the Stealth scores of the goblins to see whether the ambushers are noticed. The Perception of the characters can be lowered by the players stating that they are moving at a fast travel pace, but it is not increased by a player saying that their character is specifically looking for danger3, because it is assumed that they already are.
As you describe the party's overland travel, you might bring up various features of the landscape that could be ambush points. If any of the players say that they want to check for hidden foes, again, the answer is "you already are".
Surprise happens before you can Search
At some point, however, you will decide that the PCs are not just traveling across empty terrain, but have come upon the goblin ambush. The "Noticing Threats" rules calls this "Encountering Creatures"4, and since the goblins have hostile intent, we now need the rules for "Surprising Foes"5. This marks an important narrative and rules shift from Chapter 8 (Adventuring) to Chapter 9 (Combat).
Once your PCs are in Combat, your player's desire to "actively search" for Hidden foes is relevant - it is accomplished by their PC taking the Search action on their turn, since Searching is one of the Actions in Combat. If a PC Searches, they do make an active Perception check by rolling their Perception against the goblins' Stealth. However, a PC can only Search on their turn. Before the PC can start their first turn, the DM has to initiate the Combat by checking for Surprise and calling for all participants to roll Initiative.
The rules for Surprise6 mark the transition from Traveling to Combat. Importantly, a PC starts the Combat either surprised or not, based on their Passive Perception score6. They were already actively looking for danger, but if their Passive Perception score was not good enough to spot the worst-Hidden goblin, then they will be surprised when the Combat starts and they will not be able to do anything on their first turn, regardless of their Initiative.
If, however, the PC's Passive Perception score was good enough to notice at least one goblin, then they are not surprised by any of the goblins and start the Combat aware of the ambush. When their turn comes up in Initiative order, they will already be aware of the presence of at least one goblin and possibly more7. When the non-surprised PC's turn arrives, they can take the Search action, allowing them an active Perception check by rolling their Perception. If they do roll higher than their Passive Perception score, they may then become aware of any goblins that were previously Hidden to them at the start of their turn.
However, even if they don't take the Search action, they are still actively looking for danger8, and their Passive Perception score remains in play, potentially alerting them to the presence of goblins who have not taken their turns yet (or others who already took their turns and then Hid again).
In short, the player's desire for an active check can only be met on their own turn, but they don't have a turn until the combat starts, and by the time the combat starts they are already either surprised or not based on their Passive score.
What should you use for the Goblins' initial Stealth scores?
If just a few goblins are involved in the ambush, you can roll Stealth for each of them, when they Hide in advance of the characters' arrival. However, the more goblins there are, the more likely it is that the party will notice the ambush and not be surprised. To some extent this is realistic, because presumably a few goblins would select the best hiding spots first, while if more and more of them were involved the options for hiding places would get worse. Also, the party has only to notice the worst-hidden goblin, and in a larger group there is more likely to be a goblin with poor skill or bad luck. However, what is not realistic is the extent to which the success of the goblin ambush will steeply decline with additional goblins. This is a result of the way active Stealth is determined in the rules, as a d20 roll (representing luck) modified by the Stealth bonus of the goblins (+6, representing skill). Notice that usually the impact of the luck portion is far greater than the impact of the skill portion of the check. Whenever this is true, and when only the worst roll among many counts, we have a situation called "rolling to failure". Given enough goblins, one of them will make an unlucky Stealth roll, with a result below the Passive Perception of the PCs, who will then be alerted to all the goblins in the ambush. Against a PC with a Passive Perception of 12, a single goblin will roll a Stealth of 13 or better 70% of the time - but the chance of ten goblins all beating the Passive Perception of that PC is less than 3%. Again, if any of the goblins roll poorly, even a moderately Perceptive PC will not be surprised by any of them.
In order to handle the goblin ambush with more verisimilitude, and not use it as an exercise in "rolling to failure", you have a number of options. You could permit the goblins to to collectively make a group check for Stealth, you could allow the goblins to each use their Passive Stealth Scores of 16 as the baseline for their ambush (and only require them to start rolling Stealth actively once in combat), or, if the goblins have had at least a minute to prepare for the party and know they are coming, you could permit them "automatic success" and begin the ambush with a Stealth of 26.
Footnotes and Citations
1 'Active' and 'passive' describe the mechanics of the Check, not the kind of Perception. 'Active' and 'Passive' do not refer to what the PCs are doing. The PCs are assumed to be actively looking for danger in either case. An active Perception check is one in which the player actively makes a roll, while a Passive check is one in which the DM uses the Passive Perception score of the PC without a roll being made.
2 Examples of such activities are given as navigating, drawing a map, tracking, and foraging.
3 The assumption that the PCs are already actively looking for danger when traveling, even when the DM is using Passive Perception scores, as stated can be found in the Noticing Threats section of the PHB, which says (emphases mine):
Use the passive Wisdom (Perception) scores of the characters to determine whether anyone in the group notices a hidden threat. The DM might decide that a threat can be noticed only by characters in a particular rank. For example, as the characters are exploring a maze of tunnels, the DM might decide that only those characters in the back rank have a chance to hear or spot a stealthy creature following the group, while characters in the front and middle ranks cannot.
While traveling at a fast pace, characters take a –5 penalty to their passive Wisdom (Perception) scores to notice hidden threats.
[...]
Other Activities
Characters who turn their attention to other tasks as the group travels are not focused on watching for danger. These characters don’t contribute their passive Wisdom (Perception) scores to the group’s chance of noticing hidden threats.
4
Encountering Creatures. If the DM determines that the adventurers encounter other creatures while they’re traveling, it’s up to both groups to decide what happens next. Either group might decide to attack, initiate a conversation, run away, or wait to see what the other group does.
5
Surprising Foes. If the adventurers encounter a hostile creature or group, the DM determines whether the adventurers or their foes might be surprised when combat erupts. See the Combat section for more about surprise.
6
The DM determines who might be surprised. If neither side tries to be stealthy, they automatically notice each other. Otherwise, the DM compares the Dexterity (Stealth) checks of anyone hiding with the passive Wisdom (Perception) score of each creature on the opposing side. Any character or monster that doesn't notice a threat is surprised at the start of the encounter.
If you're surprised, you can't move or take an action on your first turn of the combat, and you can't take a reaction until that turn ends. A member of a group can be surprised even if the other members aren't.
7 The PC's Passive Perception score might have been better than the Stealth of multiple goblins at the start of the ambush, and other goblins may have revealed themselves since the start of the combat, since a goblin that started the combat Hidden from the non-surprised PC but had a higher Initiative could have attacked before the PC's turn, forcing it out of Hiding.
8 In the Using Ability Scores section, in the sidebar on Hiding, the rules describe the procedure for hiding in combat. Even if a PCs action for the round is not Search, it still gets a Passive Perception check to see Hidden enemies (emphases mine):
When you try to hide, make a Dexterity (Stealth) check. Until you are discovered or you stop hiding, that check's total is contested by the Wisdom (Perception) check of any creature that actively searches for signs of your presence...
In combat, most creatures stay alert for signs of danger all around, so if you come out of hiding and approach a creature, it usually sees you. However, under certain circumstances, the DM might allow you to stay hidden as you approach a creature that is distracted, allowing you to gain advantage on an attack roll before you are seen.
Passive Perception. When you hide, there's a chance someone will notice you even if they aren't searching. To determine whether such a creature notices you, the DM compares your Dexterity (Stealth) check with that creature's passive Wisdom (Perception) score, which equals 10 + the creature's Wisdom modifier, as well as any other bonuses or penalties. If the creature has advantage, add 5. For disadvantage, subtract 5. For example, if a 1st-level character (with a proficiency bonus of +2) has a Wisdom of 15 (a +2 modifier) and proficiency in Perception, he or she has a passive Wisdom (Perception) of 14.