A customer of mine has 1C Enterprise software running their company. They use it for managing stock, finances, employees etc.

I was asked to build an eCommerce site for them that would be always in sync with their stock and finances. I have did some research, but could not find 1C plugin/integration with popular eCommerce software like prestahop or similar.

I would like to ask what would be the quickest suggested way to go about creating a web store for them?

Hello, Thomas D!

Please see Bitrix CMS, it has built-in integration with popular 1C applications.

Hello!
The only way is to make exchange module. As I know, 1C sulutions support only 1c-Bitrix CMS.
But it is possible to write your own exchange mechanism via XML - if you need help, write me.

We have ready integrations with nopCommerce, eBay and amazon but it's part of our own configuration and it will be difficult to implement it in other configuration
Everything works online through WS and COM objects, no files exchange.

standart 1C solution have only Bitrix CMS built-in. Third side company did some plugin for another popular CMS(Drupal, Host CMS). We use some of it in our configuration

I can help with developement

I use 1C integration with woocommerce from wordpress...

You can use web services feature of 1C:Enterprise platform or simply create a PHP page that will receive a POST request, which you generate in your application.