My company doesn't have the capability to link our inventory with our online store so I need to be able to have customers place an order online, submit it without paying, we then get the order and check to see if we have everything in stock, edit the order if needed, and then send them an email with an invoice to pay for the items. Is that something that Shopify can do?
I am not fully confident you can do that with Shopify. However, you can send a draft order invocie to customer, if they pay, the actual order will be created in the backend.
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It will absolutely affects the marketing and conversion rates. If you do not have stock then just try dropshipping mode, once the clients placed order you buy and ship from supplier. If you have stock then order an ERP to upload inventory, then connect the ERP with Shopify, or stock in 3PL warehouse, they can connect inventory with Shopify.