How can we distribute Shamir's secret sharing scheme shares without a dealer? Let's say $k$ users have shares to a $(k,n)$ secret sharing scheme, and they want to issue new shares. How can they do this without collecting the shares together?

In particular, no set of $k-1$ users (including the new users) should be able to learn the shared secret during or after the share generating process. There is also no trusted third party.

One way to do this is by doing a secure multiparty computation. Is there a way to accomplish the task without a general multiparty computation scheme?
Here is the setup I am assuming, in the beginning you have a dealer who distributes $k$ shares from a $k$ out of $n$ secret sharing (e.g., Shamir) to the $k$ parties currently present.

What you would like is that if at some point in the future the $k$ parties want to add an additional party, they can do so, without requiring the dealer to cooperate at that moment in time.

As you mention, one way to do this would be with a general MPC. Another way to do this, would be, if the dealer agrees at the start that some additional shares should be available so that if the $k$ parties want to let another person in, they can, the dealer can generate a few additional shares right off the bat (say $k+t$ instead of just $k$ shares). Distribute $k$ shares to the $k$ parties, then secret share the remaining $t$ shares with the $k$ parties. That way, each of the $k$ parties has their own share, and shares of the remaining $t$ shares. If they want to give someone else a share, they send them a share of one of the $t$ shares. That party reconstructs to get a share of the original secret.

This allows the $k$ original parties to keep their original shares, while giving a new share to a new party. The dealer can also set thresholds on how many of the $k$ parties need to come together to pass out additional shares (it doesn't have to be all $k$).

As I mentioned in my comment, if you remove the requirement that the original $k$ parties can keep their original shares, i.e., you allow them to get a fresh share at the same time that you are generating a new share for the incoming party, Desmedt and Jadjodia's paper on redistributing secret shares is the way to go (malicious model variants have been proposed if necessary).