Happened to me (designer/coder/hoster) before – and will happen again. My approach always was that it's of course sad to see clients go, but of course it's ok if they take their website with them. And if that means they want to re-write the code but keep the layout I rather feel flattered, than angry.
So my point of view is that your website is your website – even if they designed and built it. It might be concidered unpolite if you'd change logo, layout, design or even changing the cms without first talking to them.
That said you should probably first try to talk to the company that created and hosted your website. Tell them you want to move and take your website with you – and ask them if they have any problem with that or doubts about it. Sooner or later they would find out anyways.
ps: naturally I would answer your question "Is rebuilding someone else's website considered plagiarism?" with 'yes'. But since it's your website things are a bit more complex. So I would recomment you leave out the "someone else's" and shorten your question to "Is rebuilding a website considered plagiarism?". The shortened question in the context you described I would answer with 'No'.