I need to write a dialogue using an formal register.

The difference between a and an is the first letter of the word that comes after it. If the first letter of the word after it is a vowel (A, E, I, O, U) you say a (a formal register, a dog, a cat). If the first letter of the word after is a consonant (every other letter) you use an (an arena, an octopus)

It’s a pleasure for me to finally connect with you in person.

Without the word with, the implication could be that he is connecting her to something else - he could connect her to another person, a carabiner, a piece of hot glue. Not that those all make sense. But we use the word "with" when using the phrase connect with in order to say that we are meeting together and sharing ideas between ourselves.

It’s a pleasure for me to finally connecmeet you in person.

I'd probably recommend getting rid of this altogether, as James has already said that it's an honour to meet Dr. Vance. As a result this feels redundant.
As a response I'd likely just write "Thank you, your work is very impressive." (combining with the next sentence)

Your work is very impressive.

See above comment

Would it be acceptable if I addressedCan I call you as James?

Please, feel free to call me JamesFeel free.

But I would be very interested in continuing our conversationhat later.

Making this more informal - conversation isn't incredibly formal, but "chat" has a more informal feel to it.

It was a genuine pleasurnice to meet you.

This again feels a bit repetitive.

If you have any questions or difficulties, please don’t hesitate to contact methere's anything you'd like to know, or need help with, give me a shout.

Again rewriting to feel less formal. "give me a shout"

I am at your disposalWhatever you need, I'll be there.

The original sentence feels fairly formal to me.

I hope you will feel comfortable throughout theLet's make this a good conference.

Again the sentence feels fairly formal to me, so I've rewritten it to feel less formal. Using volitional and active language makes it feel like James is involving himself more personally than simply wishing well on Dr. Vance.

Thank you, that i's very kind of you.

Contractions make sentences feel less formal. Even in formal settings people will contract - people who don't use contractions can sound a bit robotic.

The pleasure was all mine.

This kind of phrase feels fairly formal - I'd probably recommend removing it.

I look forward to mseeting you again soon.

The original sentence kind of sounds like they haven't actually met yet - like they were chatting online before meeting in person.

It’s a pleasure for me to finally connect with you in person.

Would it be acceptable if I addressed you as James?

This phrasing is understandable but no one talks like that even in a formal setting. You could say "may I call you James?"

Please, feel free to call me James.

Nothing wrong with this but most people will leave off the end and just say "please, feel free to" or "please, feel free"