Hi there, thank you to all for your time in correcting my notebooks. I appreciate all your unconditional support :) 

Now I am trying having my first contact with business more briefly and please advise me on how to change it in a more natural manner (how to change it to make it sound more natural). Thank you

Hi there,

I am writing to contact with the sales department of your organisation.

I would appreciate it if you could please provide me with the details necessary in order for our company to (seek corportation)* with you.

I am looking forward to hear from you. (I look forward to hearing from you.)

Kind regards

*"seek corportation" doesn't make sense mostly because corportation isn't a word. My thoughts are perhaps you meant "cooperation" (as in working together for mutual benefit). If this is the case I'd suggest, begin working OR begin doing business 

Since this is to the sales department, I imagine your company is interested in buying their product. It would probably be good to include that- that you're interested in buying product "X" for a certain time period or certain quantity.

Blue are my suggestions, pink are missing words, crossed out are incorrect.

first contact

Hi there, Thank you to all for your time in correcting my notebooks. I'm grateful for all your incondicional support :)
Now I am trying having my first contact with business more briefly and please advice me how to change it in a more natural manners. Thank you

Dear Sir or Madam,

I am writing in order to make contact with the sales department of your organisation.

I would be grateful if you could please provide me with these contact details as our company would like to work in cooperation with you.

I  look forward to hearing from you.

Yours faithfully

Hello Jasmina,

I have corrected the letter you have posted but I think you should write a little about the company you are writing from. For example...

I am writing on behalf of .... company, we specialise in/produce.. and would like to make contact....

Hope this helps