Is it ok to eat veggie hot dogs/burgers on fast days?

Of course.

Vegan meat substitutes are mostly soy and have high estrogen-like compounds.  Would advise against it ... especially if you're a man.  But otherwise quite licit.

Is it ok to eat veggie hot dogs/burgers on fast days?

Your title says vegan and your question says veggie. Not exactly the same thing. Why not just eat veggies? Much better for your health than "dogs".

on fast days we can eat meat, no?

Yes, unless it's also a day of full abstinence.  I believe that's what OP had in mind.

A lot have soy but not all. The salt content is high. There's more people who want to eat healthy and are better informed then in the past when only fish on Friday was a thing.
Immigration has increased the number of people who don't normally (plus eastern catholic/orthodox who are vegan 200 days of the year)eat meat. The stupid fat is bad for you thing also contributes as well as people suffering from cancer who awaken to the ills of processed foods.
The spirit of The fast is I think much more important for Latin catholics because your tradition is all about Lent. You have only the Lent period (obviously ember days) as specific set aside times for penance.
As St Benedict says Lent is a time to repair past negligences 

Since when did these people abstain from all animal products - meat, eggs, cheese, milk, yoghurt, honey, etc, etc, for almost 7 months every year?

The traditional fasts in the Eastern churches are:
Lent - 48 days
Phillip's fast - 40 days (from Nov 15 to Dec 24)
Dormition fast - 14 days (Aug 1 to 14)
Apostles fast - 0-6 weeks (from second Monday after Pentecost to Sts. Peter & Paul)
So that's 102-144 days of abstaining from meat or dairy, plus a week of no meat (cheesefare week). 
Traditionally, in the other weeks (30-36 weeks), Wednesday and Friday are also days of abstaining. So that's another 60-72 days.
So in total it would be some 170-200 days without meat or dairy.

That's why some advice about spirit of the fast is kinda ignored by easterners

From an orthodox priest is this snippet and kind of shows the different perspective

For Orthodox Christians, Lent and all of our ascetical effort (which includes fasting around half the days of the year, not just Lent, as well as other ascetical practices, none of which include pickingsomething to give up) are about neither the Pietistic emphasis on self-discipline and individual piety and belief nor in the more Lutheran concern of revealing us to be sinners (thus representing the continuum between the Radical and Magisterial Reformations, respectively). There is nothing wrong with either goal, of course, but that is not the point of asceticism