I presume the term “Brexit implementation period” means implementation of the negotiated deal with the EU.
Does this mean the “implementation period” is predicated on a deal with the EU?
If so, what happens on March 29th if the deal is voted down?
Any “implementation” period is certainly predicated on there being a deal of some sort. How you want to call that period does not matter as much as the fact that there is no legal basis for anything else than the hardest of Brexit otherwise. Without additional agreement, article 50(3) would automatically apply and Brexit would be fully effective immediately.
Conceivably, this deal could be something else than a fully fledged trade and cooperation agreement and include things to be defined later, either through a follow-up agreement or at a lower level through some sort of coordination mechanism. If that's the case, the “implementation” or “transition” period would actually be an extension of the negotiation period. Whether the EU would be amenable to that and agree to fudge the nature of the delay by pretending it's about mere “implementation” is another question…
But whatever the contents of that agreement, it has to be approved by EU institutions and national parliaments, very much including the Commons. If it was voted down by any of these parliaments, there would not be any “implementation” period without another deal, it's as simple as that.
Beyond that, I don't think anybody knows how it would play out. I would expect some emergency meetings to find some 11th hour promise or declaration that would make another vote possible without touching the letter of the agreement itself or perhaps all EU member states agreeing to extend the article 50 deadline to find some time to hash out another agreement (as the treaty itself provides). Those are the two ways I can think of to prevent the UK from crashing out of the EU, on a purely formal/legal level. Again, what it would mean for the actual substance of the agreement and UK/EU relationship is another (very speculative) question.
In a way the rules are really simple: Two years after an EU member state delivers an article 50 notification, it is out of the EU unless there is a negotiated agreement to do something else. What this "something else" will be is up to the states.