Yes she can leave on a foreign passport, but I'm not sure how smooth or bumpy her exit from Estonia will be (as there's a division who can and cannot be a dual citizen).
As your wife is Estonian, your daughter is therefore Estonian citizen by birth (not due to her being born in Estonia).
The Citizenship Act states
§ 5. Acquisition of Estonian citizenship by birth
(1) Estonian citizenship is acquired by birth by:
° 1) any child at least one of whose parents holds Estonian citizenship at the time of the birth of the child;
Now, even though Estonia does not allow dual citizenship the same Citizenship Act says:
(3) No one may be deprived of an Estonian citizenship acquired by birth.
Meaning, she can lawfully be a dual Estonian citizen (she's not unique in this case, other people are legally dual Estonians too, even the previous president of Estonia Toomas Hendrik Ilves was a dual US-Estonia citizen until he decided to give up his US citizenship himself).
Now, I'm not very familiar with Australian law, but if it's any similar to the British Nationality Act, then it might be the case that your daughter is already an Australian citizen and you don't need to apply for the citizenship per se, but rather can directly apply for her Australian passport.
Saying all that, I strongly believe she should not have a problem leaving Estonia by either Estonian passport or Australian passport. If the question would to arise on why her Australian passport has no mention of entry into Estonia you should be able to confidently say that she is Estonian (as an Estonian by birth she is protected by constitution and cannot lose her citizenship just because she has another one).
But to make things easier, in case she would ever travel back to Estonia, I would also get her an Estonian passport. She can board the plane to Estonia on her Australian passport but I wouldn't personally want to explain to Estonian immigration she is Estonian while waving her Australian passport. Even though they should let her through, having both passports at hand might help speed up entry/exit check at Estonia in this particular circumstance.