Failing solo Vaporeon
I'm trying to solo the T3 raids and have succeeded (and tried) only Machamp. Last time I failed Vaporeon and timed out in red. So I decided to power up some of my team. Now I failed with no hp left, but still timed out twice. I managed to dodge once to get an extra SoB in. The battle started at 177 seconds and even ended at 1 or 2 seconds left. The team I used:
Exeggutor BS/SoB 14 att LV 33
Exeggutor E/SoB 15 att LV 33
Venasaur VW/SoB 15 att LV 31.5
Jolteon VS/TB 15 att LV 30
Dragonite DT/HU 15 att LV 30
Jolteon VS/TB 14 att lv 30
What am I doing wrong, Vaporeon is supposed to be easily soloed?

Answers
Because Weezing (and Muk) have a lot of possible moves (dark and bug) that its Psychic type counters are weak to, and said counters are rather glassy. Exeggutor on the other hand has no such moves and even has a double weakness to bug, additionally, its counters tend to be tankier and still very powerful.
Each tier raid bosses have a fixed HP value, which mean defense is the only stat that judges the difficulty of the raid. Vaporeon has only 177 in base defense, while Jolteon and Flareon have 201 and 204 respectively, making them relatively tanky within the same tier. That's why Jolteon and Flareon are (nearly) impossible to solo.
I use a fully maxed. Raikou, Exeggutor, Jolteon, and Zapdos. My best time had almost 30 seconds left. My worst was 8. I think your team is grossly under leveled and while I like Voltswitch I would recommend thundershock for this instance. Never needed the 5th slot. I tank aquatail mostly because it just keeps coming and coming and coming. I dodge pulse and pump. Good luck!
At close to level 30 Vaporeon raids are nearly always a race against the clock more than survival. So that being said, your first three team members are high enough level with the right movesets. The last three all have movesets that are too slow. To be successful you'd have to switch VS to TS and try to find another High level high attack eggy with SoB to replace the dragon.
Here's some suggestions with your current squad.
BS Exeggutor - Bullet Seed's next damage breakpoint (10) with 14 attack IV is at level 35
ES Exeggutor - Extrasensory has already reached max damage (10) so powering up this one has little difference on raid outcome, either keep it as is or TM it to Bullet Seed and power it up to Bullet Seed's next breakpoint which is level 34.5 with 15 attack IV.
At that point the only difference between Bullet Seed and Extrasensory is in energy per use, 14 vs 12. Both can only reach a maximum of 10 damage.
Venusaur - keep it as is, it's got a damage breakpoint at level 39 that only 15 attack IV can reach but obviously that's not worth it just to solo Vaporeons.
Jolteons - TM them to Thunder Shock. They have better dodging and give more energy, plus slightly more DPS currently - Thunder Shock reaches it's maximum of 6 damage per use very early, (6 damage / 0.6s = 10 DPS) Volt Switch doesn't hit 23 damage until level 32 (23 damage / 2.3s = 10 DPS), but even at level 32 Thunder Shock is still better for the reasons listed earlier.
Dragonite - is a bit slow even with it's best simulated moveset, Dragon Tail + Outrage, clocking in at 184s. If you have another good grass or electric, swap it out because they'll be faster, but if not it should still be fine to use because that expected time of 184s assumes you're only using Dragonites, which you aren't.
I don't think you need to power anything up at all. I would tm one Jolteon to Thundershock and lead with it. You want TS because with the electrics, you really need to dodge one and only one charge move, regardless of what it is. Then bring the grass and don't dodge anything. I would send the other jolteon with either TS or VS after that and try to finish it with a final TB. Stick your Dragon at the end because I highly doubt you'll get that far.
If that fails, find yourself a Raikou and/or Zapdos, power them up to level 30 and go to town.