So I am doing an Ironman play of the game, meaning I am trying to roll with the punches of whatever happens. Gurke approached me in the bar and I assumed he was another assassin and my party started attacking him.

As it turned out he isn't but now officer Vai and her flaming fistians are out for my blood. I tried going to the nearby church and donating a little but it did nothing. Now they try to attack me every time I enter and I am trying to traverse the game without killing them. Is there any way to get forgiveness? Namely I want to make Officer Vai not an enemy anymore. My reputation is 14. I did not kill anybody, just did minor damage to Gurke and we ran.

Considering that you're doing the Ironman run, the only option is to charm person who you don't want to be hostile anymore. Normally you would simply reload, because there isn't any other option to make people neutral once they became hostile.

I don't know if you're into cheesing, but in Beregost there's a cloak that let's you charm people pretty much without a limit. You might consider this if the charm person (or mammal should work too) is not enough. Once a character is charmed you can talk to them by using the conversation button and them clicking at the character that you want to talk to.

Interestingly enough there's quite a bit of additional dialogue for charmed NPCs in the game.

when charm wears out they turn red again... just be awear of that

you can do good for another point of rep and come back later ... killing offier V is not a great idea

This is what I was looking for- a more permanent solution. I have already charmed her once as I had Xan in my party and my character is Chaotic good and I am not looking to start murdering everybody now. I hoped a reputation point would help.

1 i'll assume you are at least chapter3 - you only need confront Tranzig on the second floor. to unlock the bandit camp [different inn from officer V] so my advice is move on with the plot before you make things worse

2 charm stops dialog so using it on any npc you need to talk to will break the game... however there are tricks like pickpocketing that rely on it

3 next run talk to to the Bard... his side quest is worth doing just for the good/evil choice

Then Algernon's cloak is probably your only option that I can think of right now. Algernon is at the first floor of the Feldpost Inn, you need to steal the clock from him and considering the Ironman run, I would suggest using a thieving potion, to increase the chance of stealing (I assume that you don't want to kill the guy).

This cloak lets you charm people pretty much with no limit. The characters will remain charmed for a very long time (I would say permanently, but there probably is a time limit, just very long).

No reputation increase will stop characters from being hostile unfortunately. Once they turn against you, charm seems to be your only option (besides cheats).

Algernon's cloak is actually a quest item, but the quest itself starts in the Baldur's Gate city. It's quite a cheesy item, due to its power (or almost no limits I would say), so I only suggest to use it because of your situation.

You can talk to charmed characters, you just have to use the conversation button to do it.

Well, thats why I think stuff like Ironman mode or Honor mode are silly ideas.

If you run into a bug, you cant fix anything. Unless you manage some serious console magic.

Unless you are actually a sane person and keep backups of your games. Because you actually think the time spent on getting these saves is something that shouldnt be lost. But then ... that circumvents the very idea of Ironman/Honor mode. Hu !

Charm only stops dialog in BG2, in BG1 in the original you could talk to charmed NPCs, and there was quite a bit of special dialog from talking to charmed NPCs, including some foreshadowing of Irenicus where you can talk to his lover who was transformed into a monster ala Medusa.
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Also the Nymph Cloak works as a substitute for Algernon's Cloak, but is inferior.
Algernon's Cloak gives you 65535 charges of Charm Monster.
The Nympth's Cloak gives you 32767 charges of a variant of Charm Person, but has a MUCH shorter range that is pretty much touch range, so you wind up taking hits.
This was modified in EE, I think both have a fixed number of charges per day in EE.

yes and no
yes bg1 original had lots of different rules/ code to bg2 original and yes EE does use all the same code for both Bg1 & bg2 but they only followed on what modders had already done i.e, even without EE its common for mods to play bg1 using the bg2 rules so everything works the same throughout your adventure