I'm looking for a Latin equivalent to the personification 'Mother Earth'. Does Matris Orbis or Mater Terram do the job? Do they translate in the same way?
I propose Mater Terra or Mater Tellus. See online Latin dictionaries to see which word for "Earth" is more suitable. If you mean the planet as a whole, I suggest Tellus. If you mean the ground under our feet, I suggest Terra. The most suitable nuance depends on context.
The Latin word for "Earth", whatever you choose, should be in nominative. Here "Earth" behaves like a given name. Thus Orbis would be grammatically fine but I doubt it means what you want.
Edit: The Latin word order is quite free and I chose to put Mater first because it sounded best to me. It seems that my intuition was wrong, as the examples in sumelic's answer demonstrate. Therefore Terra Mater or Tellus Mater is better. The word order I originally proposed is not wrong as far as I know, but it seems less idiomatic.
I see two approaches here: the literal and the historical one.
From a semantic point of view, your choice of words is mainly right, but as Joonas points, orbis means primarily something round, and its assocaition (and meaning change into European languages) with the whole World derives from the fact that its use with terrarum and terrae (the roundness of the Earth, the round Earth) is rather common.
Regarding the grammar, Latin noun endings change according to the function they fulfill in a sentece, hence:
is the way to put it as a stand alone name, in an English text, or as the subject of a sentence, but bear in mind that this might change if you want to go further. For example, if it is the object of the action, in Latin you would rather say/write Matrem Terram, if you are talking about something else that belongs to Mother Earth you would go Matris Terrae, etc.
In case you want to stick to Roman Mythology, then Mother Earth as a godess actually has a proper noun:
I agree with Joonas Ilmavirta's answer in that you want to use tellus or terra along with mater. But I am not sure which word order is more idiomatic in Latin. (It's true Latin word order is relatively "free," but there are still tendencies towards using certain orders in set phrases.)
Tellus mater teque Iuppiter obtestor.
I believe this is an example of the phrase "tellus mater" being used as a single noun phrase meaning "mother earth"; however I am not a fluent reader of Latin so my analysis of the grammar of this sentence may be incorrect.
Some other relevant results I found using this corpus:
primum, qui omnis fructos agri culturae caelo et terra continent, Iovem et Tellurem: itaque, quod ii parentes, magni dicuntur, Iuppiter pater appellatur, Tellus terra mater. (Rerum Rusticarum de Agri Cultura, Liber Primus, Marcus Terentius Varro)
- seems to use "terra mater" as a title for Tellus.
Terra mater est in medio quasi ovum 15.1 corrotundata, et omnia bona in se habet tamquam favus'. (Petronius, Satyrica 39.1.1)
- seems to use "terra mater" as a poetic term for the Earth.
I was not able to find any results when I searched for "mater terra" or "mater tellus" in this corpus.