I'm not sure why you thought you should stand a chance in this scenario - it's 3v1 and each of those three eagles is much faster and slightly more maneuverable than your vulture - and your large hardpoints offer very little benefit against eagles because they have no hull hardness in the first place. Combat in this game is about more than just dps and shield strength.
Essentially you were being shot at by NINE hardpoints and you only brought two to the fight. The Vulture is a "big ship killer" that is compromised against smaller ships so in this case it was a bit of bad luck - both in the 3 vs 1 situation, but also in the fact that slow vultures are easy prey for jousting eagles and you ended up bringing scissors to a rock fight. Rail guns are extremely common on smaller ships because they are suited to the higher maneuverability and offer good piercing to compensate for the drawbacks of smaller hardpoints.
You can't win every encounter because the encounters aren't always fair. Knowing when you are in trouble and needing to run (3 eagles would trouble even an FDS or FAS, most likely) is an often-overlooked part of combat. Fly safe CMDR o7
On the other hand, Eagles fold like tissue paper once their shields are down. A typical NPC Eagle can be destroyed by about five or six seconds of sustained fire from a single gimballed beam laser. The scenario described here is very similar to one of the ones off the training menu except the Eagles are armed with lasers and, well, I have no idea what rating the modules on the Vulture are. I've never checked. You do get one gimballed beam laser and one gimballed class 2 multicannon. I suspect the laser is a class 2 as well.
I think the training scenarios are pretty good practice, especially the Incursion ones where you can pick the rating of your NPC opponents. There are four levels - Mostly Harmless, Competent, Dangerous and Elite. I've only tried out the two lower levels so far. I don't consider myself a super pilot in any way, probably competent if anything, but I haven't lost my shields on ether of those scenarios to date. They're a great place to practice against NPCs, one on one. I think there are many on one scenarios too, but I'm not sure. Well, there's at least a two on one, and a three on one scenario. Anyway, they're a great place to try out boost turns, FA-off stuff, circle strafing or other fancy flying without facing a rebuy. The Mostly Harmless level opponents are somewhat like flying potatoes that even my nine year old son can destroy. He is beaten by the Competent NPCs though, but perhaps that will change soon with practice.
Sure, you get given a set ship but it's not all bad - Mostly Harmless gives you an Eagle armed with two fixed burst lasers (hate) and one gimballed multicannon. Competent gives you a Viper Mk III with two beam lasers and two multicannons, all gimballed. That's where I discovered that it's actually a really nice combat ship to fly - nice strong shields and decent firepower. I don't know what ships the other two levels give.
However in one of the other challenge scenarios, you get an Asp Explorer with two class 2 beams, 2 class 1 beams and 2 homing missile launchers. You start off against Eagles and Sidewinders and the end NPC opponent is an Anaconda. No reloads or repairs. If you can beat that, I reckon you've got the AI nailed, and you're a better pilot than I
Best I've done is taken the Anaconda down to about 10% hull before ”eject, eject". My twelve year old son gets to the Anaconda too...