Yes, No, Maybe. All of those, depending on where they want to buy it.
When buying property in a country, you generally have to do it under the jurisdiction of the country where the property is located. That means which flavor of contract law is involved varies wildly.
In general, though, contracts can be offered & accepted both verbally (oral) or in written form. An verbal contract is, in most jurisdictions, as valid as a written one.
The usual elements of a contract are those:
- Offer and acceptance
- Lawful purpose
- Lawful consideration
- Certainty and completeness of terms
- Free consent of the parties
Lawdepot on Verbal Contracts
So, can we do all of these? Well, astronauts can communicate with earth, so an offer can be extended and accepted. This can be both verbally as well as n writing. Buying real estate is (presumably) a lawful purpose. We assume that the astronaut is willing to pay cash & has means to access & transfer funds.We can assume that astronauts can access their online-banking accounts. Certainty and completeness of terms? Its certainly possible. Capacity? Yes, astronauts are usually mentally capable of entering contracts.
So in theory, nothing prohibits astronauts from entering contracts, at least not in most jurisdictions (I'm not scouring contract law of about 200 countries).
But there is a big but. Not all contracts can be done verbally or without the proper written form. In the US "According to the statute of frauds laws for most states, the following types of contracts must be in writing and signed in order to be valid: Contracts for the transfer or sale of land [...]" Legal Match / Contract Writing Requirements. Other countries, for example Germany, have similar restrictions. Thus, shipping a contract to the ISS, signing it & shipping it back might be required to legally purchase property.
One might think that this is the end of all considerations, but in fact it isn't. The world is a big place, and jurisdiction vary. I'll only slightly tough on one country I'm familiar with, Germany. In Germany, land ownership is registered in the "Grundbuch", or land register. This register says who owns which piece of land.
The following is some hefty legalese:
You should know the principle of abstraction. This states that a change of ownership requires both a commitment and a disposal transaction. You should expect two separate transactions for this: The purchase contract represents the commitment transaction, while the actual transfer in the land register corresponds to the disposal transaction, that is, the transfer of ownership. However, here too, the conveyance must be declared by an authorized body, and this is usually done by a notary.
Source (in German)
In practice this means that buying a plot of land without a notary is pretty much impossible unless you are very knowledgeable about the laws involved, at least in Germany.
So to answer your question, its a solid: It depends. Regulations are different from state to state even in the US, and the question wasn't even restricted to the US (we do get astronaut, cosmonauts & taikonauts from a variety of countries, who could all try to buy property in their home country or even abroad). It might be possible in some states/countries to buy land online or by phone (verbally), it might require shipping written contracts to the ISS for signing, or it might -- in practice -- be impossible due to the legal requirements surrounding sales of real estate in some jurisdictions (e.g. Germany).