According to copyright laws, the artist has the right to sell reprints of the art.
It is important to know that copyright nearly always rests with the
artist, regardless of who owns the artwork. There are exceptions to
this rule, such as work that has been specifically commissioned or
completed during employment, in which case copyright stays with the
commissioner or employer. Freelance artists working for a range of
companies should keep this in mind.
However the copyright can be sold.
You need to be 100% clear what type of permission to print you have
sold, and what restrictions on future commercial activity the sale
might entail. If you do not understand the terminology used in the
contract or agreement, you must not be frightened to ask for an
explanation of its implications.
There are three common categories of sale for copyright and
Artists sell copyright outright. They have no control over the ways in
which images are subsequently used, and the new owner of the copyright
is free to sell licences as they see fit and to retain all the
profits. (Sometimes, however, agreement is reached that copyright will
be sold but royalties will still be paid to the artist. See below.)
Artists sell copyright for a specific limited purpose; neither artist
nor publisher can use the image for any other purpose. For example, if
the image is published as a limited-edition print it can never be used
for anything else, either by the artist or the print publisher.
Artists sell reproduction rights, or a licence, for a specific limited
purpose; the artist retains copyright and can continue to profit from
it. For example, the artist sells the image for use on a set of table
mats, but can then go on to sell it for other uses.
It is essential that you are clear whether or not you can sell licences in the > future, whether you have signed the right to do this over to someone else or
whether neither party can do so (e.g., for most limited editions).
You may be thinking that the phrase "There are exceptions to this rule, such as work that has been specifically commissioned or completed during employment, in which case copyright stays with the commissioner or employer." means that you are the employer and get the copyright. Unfortunately this apparently relates mostly to companies that have the artist as an employee or under special contracts.
Basically, from the moment something is fixed in a medium (be it a
canvas, on a piece of paper, in a photograph, or in a digital file),
it is copyrighted to the person who created it. That would be your
artist. Even if you’ve given them the idea for the painting, the
execution of the final product is copyrighted to them. Copyright
gives the artist the right to decide who can use their work and how,
whether or not it can be duplicated or reproduced, and where and how
it can be displayed.
From what I've read about reprints by artists, most of them will say at the start that they'll sell reprints later on, or ask permission to do so. It's considered common courtesy so the buyer knows what they're getting. For more money many will sell the copyright to the customer, or in the case of personal pictures of family members agree not to sell any reprints out of respect for the personal nature of the art.
Since you and the artist didn't discuss copyright issues at the start, this leaves the ball in the artists court. Maybe you can negotiate a new contract with them, but from my understanding it doesn't look good for you.
Hopefully someone with more knowledge can find a loophole or something I missed that can help you.