This page has been added to describe why multisite setups - where several sites handles one webservice account - will never work properly. Please be aware that this is not the same solution as if you were running WPML (the multilingual plugin).
First of all, the problem does not actually lies within the plugin but in Resurs Bank - it is actually Resurs Bank that does not support multisite accounts. There are some issues that is simply not supported here:
Order ID numbering
Normally, the plugin are connecting an order to its order ID in the platform by a datestamp. You can also run the plugin by setting the order id as the incremental post ID for the multisite. If you tend to follow an incremental id and initialize your first site with order number sequence 1000, your second with 2000, and so on, those order numbers will at some point in time collide.
When orders are created like this on Resurs Bank where the same webservice account are used over several sites, your sites may be able to sort this issue out. On Resurs Bank this won't happen as all orders are created on the same account. When your first site reaches 1999 and tries to switch over to sequence 2000, the webservice will stop working as an exception will be thrown at bookPayment level - saying that order id 2000 already exists. You could of course use a higher sequence as this is only an example. However, at some point - mentioned above - there might be a risk of conflicts between your sites.
Callbacks: Only one site will be able to handle callbacks as the database tables are split up to be running each site separately
When the plugin are registering callbacks it is very site specific. In a WP Network only one URL will be registered without the ability to identify which site the callback is intended to land on. In theory you may make this work by handling each callback by its order id. However, this is not recommended (see the section about Order ID numbering).
The callback URLs is also a problem
When callbacks are registered, salt keys are also registered at the webservice endpoint. Those salt keys are stored at the site where they were registered. By means, if you have four sites and register callback with site 2, the other sites will be unable to handle the callbacks due to the different saltkeying. Besides, your site 2 may register different URL's compared to your other sites. This is a limitation that resides at Resurs Bank.
What can we do?
If you feel this is a problem and really want to take the risk running one account over several sites, you can either look into the problem your self. This version is very adaptive and it may not be impossible to wrap an externa plugin around this problem by using one of the many filters/hooks. Besides, if you decide to run the plugin from a "internal orderline perspective", where your plugin is not dependent on the incremental post-id's created by WooCommerce but instead the "long strings" - you could make this work as long as the payment id can be linked to a proper order in a proper site instance.
The problem we stand with here is that the plugin must have connection with the root network site and know what each site is doing (since the WordPress Network splits up each site in individual tables). It should however not be entirely impossible, but the recommendation is to handle those parts with an own setup. If you feel you want to try that out and you seem to miss important components in the plugin it self, you can create a pull request with a suggested solution for where filters/hooks can be added for this to work. The alternative is to create an issue in one of the public issue repositories and wait for an action - which eventually can be delayed.