Usually your frequent flyer number and status level prints out on the boarding pass of most (all) OW airlines, assuming you are using the appropriate number and program on that reservation.
Note that some lounges have capacity control limits and other requirements that may limit access in some circumstances, but the physical card is rarely requested.
Also, many airline web sites allow you to print a paper copy of your card or as a last resort print a screen shot of one of the pages of your account record that shows your current status.
In short, as others have indicated, there are numerous work arounds if you do not have the physical card.
Enjoy the lounges on your trip!!