For example 'unicorn' has no extension, but we can still provide a definition of one as long as we assume that they do exist. So is it possible to define a term that really has no extension, without assuming that it does?
One felt intuition about defintions is that they be "conservative"- they should not allow you to prove anything more than you already could. In mathematics, this means that you cannot define things into existence. In natural language, which already assumes a free logic, this is fine and (in fact, as @Bumble notes) a commonly accepted practice.
In fact, natural language allows many things that a more restricted language might not otherwise tolerate, include ostensive definitions. These definitions might not fit the more "mathematical" character of other definitions, but roughly speaking they share the illocutionary force as their mathematical pronouncements. Which brings us to another point- one must distinguish between the speech act of defining and the definition itself. So it is easy to get confused, but which the author means is typically made clear via context.
for more, see SEP on definitions.
Is it possible to define a term that has no extension, without presupposing the existence of members?
Words do not have extension in themselves. Rather, we think of them as having extension or not. Thus, extension is a function of what we use the word to mean.
The phrase "the Unicorn" may or may not have an extension depending on what we use it to mean. Presumably, some people in the Middle-Ages really believed the Unicorn existed in the real world and used the phrase to refer to something real.
Today, the phrase seems to be used exclusively to refer to an imaginary being. An imaginary being is an idea in our mind and so not something in the real world, but the phrase still has extension since it refers to an idea, which is something in our mind, so the phrase refers to something that exists in our mind.
So the question is whether we can use the phrase "the Unicorn" so that it would have no extension. As far as I can tell, nobody does.
Using a phrase so that it has no extension comes down to use it to refer to nothing. That is to say, so that it does not refer to anything. One word obviously fulfils this requirement, namely, the word "nothing". In one sense of the word, it is used to refer to nothing, that is, not to refer to anything, that is, to have no extension. For example: There is nothing in the fridge.
So although it is still possible to use the word "nothing" to mean something that would somehow not exist, the word is mostly used to mean "no thing" rather than "some thing".
It seems that the word "nothing" and its synonyms (nil, naught etc.) are in English the only nouns available to achieve no extension. However, we can also use a negative to achieve the same result. For example, in the sentence "There is no apple in the fridge", the phrase "no apple in the fridge" refers to nothing, that is, it does not refer to anything.
We can achieve the same result by saying that the intersection Ax ⋂ Fx is empty, that is to say, that the intersection Ax ⋂ Fx has no extension. There are plenty of apples (A) and there is a particular fridge (F), but there is no apple in that fridge. Thus, when we say "There is nothing there", we literally mean that the intersection Sx ⋂ Tx between the set of those things that we are looking for and the set of those locations we are looking at is empty. So the word "nothing" is normally used to mean an empty intersection, which refers to nothing because it has by definition no extension.
As the sentence "There is no apple in the fridge" has the same logic as the sentence "There is nothing in the fridge", we can also interpret the latter as an empty intersection Ax ⋂ Fx between all things (Ax) and the things which are inside the fridge (Fx).