My experience is mostly related to Open Source ERP systems (and eCommerce systems).

The two options you have are:

1. Go with an ERP system that has inbuilt eCommerce.

There are several ERP systems which have this feature. Netsuite is a popular commercial cloud based system that has a built-in eCommerce system. Tryton and Nereid is an Open source alternative (Disclaimer: I work for a company that works on this stack).

At advantages of an integrated system are:

- One database to manage everything from sales to gift cards to refunds to shipping
- Order updates reflect real time across the customers my account page and your ERP.
- No more asking for credit card information when customers want to change orders.
- Shipping (fulfilment) status reflects realtime
- Pricing updates have to be done only at one place
- Possibility to have additional sales channels like a point of sale, a marketplace etc.

2. Go for an eCommerce system that has known integrations with ERP systems.

This is a popular approach too and there is a large plugin ecosystem with most ERP systems that provide integrations. The tricky aspect however is that no integration can be a 100% tight. So you need to analyze if the integration covers the features critical to your workflow. For example, most integrations provide importing of orders, but if your business uses gift cards extensively, you might want to make sure that the integration handles gift card and related accounting well.

A few disadvantages of the approach is:

- Maintaining integrations when either of the softwares change is costly.
- Introducing a feature on either systems will need development on the other. For example, if you want to use reward points on webstore, your ERP might need it too.

I will be happy to take a call to understand your use case.