I think you may be confusing this a bit. I am not aware of a "members trade first" requirement, and in any case, private entities are free to trade with whomever they wish. There is however a different requirement which might fit the argument you are citing.
What the EU requires is that all member states are treated equally in government tenders. So, if a small country wants to build a new highway, it cannot restrict the tender to domestic responders only. It is obliged to let any EU construction firm respond, and to choose the bid with the lowest total sum. What happens in practice is that established firms from rich countries (think Strabag, or frequently Siemens if it is not a highway but an automation related project) submit bids they could never fulfil if they were to bring their own personnel over, they win the bid, and then get the local firms (who would have won the bid otherwise) as subcontractors, paying local wages to almost everybody except to 1-2 high-level managers from the own firm.
This practice has a variety of effects. On the positive side, the winning firm frequently is better organized than the local ones, and if things go awry, it is a serios outfit with money behind it, and if sued, would have the money to pay. (It has the money for good lawyers too, though). Also, in small and poor countries, sometimes the local companies are in debt (sometimes through no fault of their own, if their partners frequently default on them) and are barred from bidding on government tenders. As subcontractors, they at least get the chance to work on these projects, and its workers get wages which stay in the local economy.
On the flip side, you also have a situation where the larger cut of the profits goes to the bid winner and not to its subcontractors, thus leaving the country. There are also smaller effects, such as having the job openings for the most qualified and highly paid positions in the large company outside the country, indirectly increasing the brain drain towards the rich countries.
What is most visible, and people complain about, is that "the Germans gets to build our highway now, not our guys" (they frequently don't even know about the local subcontractors, since it is the Strabag signs you see at the construction site) while they expect that, without the EU rules, the country would be able to enact a protectionist law where only local firm can bid on the juiciest projects. It is probable that you heard a version of this, and without knowing the context, you took it to mean that there is some kind of trade obligation.