There are experimental vaccines that deliver DNA of interest (e.g. that would express spike protein) merely tied to a plasmid encapsulated in a LNP (liquid nano-particle). Some animal trials indicate that LNP-encapsulated plasmid-DNA vaccines would achieve approximately 10-20 times better delivery efficiency than "naked" plasmid-DNA injection, i.e. LNP encapsulation would lead to that 10-20 times less active substance (and possibly fewer booster doses as well).
The recently approved (in India) ZyCoV-D, which is a plasmid-DNA vaccine (exact plasmid used is pVAX-1, which has been commercially available for at least a decade), doesn't seem use a LNP encapsulation though. ZyCoV-D requires three doses as the initial schedule, so delivery efficiency (via a jet injector) is probably not as good as what could achieve with additional LNP encapsulation.
Basically, a plasmid is circular piece of DNA (typically found in bacteria) that if it makes it into the nucleus, results in mRNA being produced from a certain part of it--part which follows the promoter region/gene. pVAX-1 [for instance] uses the promoter from the cytomegalovirus (CMV). CREB binding sites in this promoter region facilitate movement to the nucleus, although that might not be whole story of how it achieves nuclear import. The current plasmid-based vaccines seem to rely on rather inefficient means of nuclear import. Something like 1-3% of plasmids injected into cytoplasm make it into the nucleus.(The full CMV appears substantially more complicated in this regard, with additional nuclear localization sites in other large proteins, only more recently identified.)
LNP encapsulation is a fairly expensive procedure, which requires investing in highly specialized microfluidics equipment, so there's probably a commercial tradeoff between "brute forcing" the solution with more DNA vs. encapsulating in LNP.
For mRNA the benefits of LNP encapsulation are perhaps even higher; one paper reported 40-fold increase in expression with LNP encapsulation than without.
I'm not totally sure about this, but currently it seems that plasmid-DNA vaccines cannot be made by the mostly synthetic route that mRNA vaccines use, i.e. plasmids still need to be grown in bacteria inside bioreactors, which makes them less attractive than mRNA-LNP vaccines in that regard.
Adenovirus vectored vaccines (like Astra Zeneca's) are grown with the virus infecting actual cells, so you cannot easily replace the DNA inside this DNA-based virus (family) with RNA. The "trick" with an adenovirus-vectored vaccine is that it doesn't replicate in most cells, but it does replicate in the ones in which it is actually grown; these cells are themselves modified to actually supply the missing replication gene for the virus.
Adenovirus-based vectors typically have two regions of the virus genome removed, known as E1 and E3. The E1 region contains early genes required to trigger a transcription cascade enabling viral replication; E1-deleted vectors therefore need to be grown in E1 trans-complementing cell lines such as HEK293 cells. HEK293 cells have a 4-kbp region of human adenovirus type 5 (HuAd5) integrated into the cellular genome that provides the E1 genes in trans enabling efficient virus vector replication and recombinant virus production. [...] Usually, the transgene to be expressed [e.g. spike] is inserted into the virus genome in place of the E1 region under the control of a highly active promoter.
There is actually a small risk that some of the viruses produced this way are replication
competent, by including (via recombination events) the original E1 gene from the cell in which they are grown. One of the reasons for selecting a virus that is not adapted to humans is to minimize the chance of bad outcomes stemming from this risk. (The other/major reason is to limit the likelihood the immune response will react to the vector before it can deliver is transgene payload.)
There are also some further similarities between the ZyCoV-D vaccine and the AZ vaccine: both use the CMV promoter to mark the start of the transgene, which is also terminated by the same BGH poly-A sequence. So, on this level, they rely on the same cookbook once in the nucleus.
Finally, RNA viruses have also been tried as viral vectors. There were even some head-to-head comparisons with DNA vectored-vaccines; like between ChAd3 and rVSV viral vectors for Ebola (2017); as far as I can tell, there wasn't much difference in either safety or effectiveness in that trial. The RNA-virus-vectored vaccine for Ebola was approved by the FDA in 2020. There's also an experimental Covid-19 vaccine based on the same rVSV platform. I'm not entirely sure why this was not pursued, but I suspect that it's because RNA-viral-vectored vaccines need -60 degrees C storage, so the DNA-based vaccines definitely have an advantage here.