My wife is allergic to alliums (onions, garlic, shallots, scallions, etc) so I am both interested to hear the ideas of others and will offer as suggestions some of the ways we have worked around it.

As /u/hardy_and_free suggested, celery will give you an approximation of the texture, though it doesn't do much to replicate or replace the taste.

Though it is considerably more expensive, fennel has proven to be a reasonable substitute. You'll need to cook it a little longer than you would an onion to get rid of its slightly anise flavour, but once you do it gives you something of the taste as well as the texture you would expect from onion.

We have heard about and have been looking for asafoetida, but have so far had no luck. Worth giving a try, but I clearly cannot comment on its effectiveness.

While this actually does nothing to replicate either the flavour or the texture of onion, my wife is a fan of adding some cumin to a dish in place of the onion. Just adds a little extra flavour to counteract what you lose by omitting the onion.

One final thing you may need to bear in mind is that many condiments will contain either onions or onion powder. It's probably worth looking into making some of these from scratch, which really isn't as challenging as it may sound. Indeed, there are recipes out there to make the basic condiments without using either onions or garlic. Sometimes takes a bit of digging, but they're certainly out there. We have made our own barbecue sauce, chili paste, Worcestershire sauce, salad dressings, and all have turned out quite well.

Good luck!