Can wormholes exist without affecting the universal energy density? The universe's shape of the universe and reflects the fate and energy density of the universe, and scientific trials have determined our universe to be flat. 
However, wormholes supposedly warp space-time on a large scale in order to connect two distant points.
Can wormholes exist if they drastically change the universe's shape?
 A: If they exist, and that is a big if that many people except science fiction writers doubt, they do not change the shape of the universe, but instead may create local shortcuts (i.e., a path through the wormhole) between two spatial areas an otherwise unaffected universe. 
The universe is most likely flat or pretty close, so wormholes maybe long effects in say 2 local areas, but the overall geometry would not be necessarily affected. The wormholes could just have the mass equivalent of a black hole (more or less), and so be as small as 3 solar masses or maybe as big as galactic centers, but still pretty small on a cosmological scale. 
Wormhole solutions exist in General Relativity, such as the Einstein Rosen bridge which arises out of an analytical continuation of a standard Schwarzschild black hole into the other side. See for example the wiki article on wormholes and its references at https://en.m.wikipedia.org/wiki/Wormhole
The problem with those and other wormhole type solutions found is that they require either exotic matter to keep the bridge open, or it closes faster than you can go through. There are variations of what kind of exotic matter, and other kinds of fields possibly, but usually it has required a negative energy density or so other properties for the matter or field. People keep looking so there may be some less out of this world possibilities, but all pretty unusual characteristics. 
Could there be some such exotic material that was created early after the Big Bang, and even through inflation somehow it managed to collapse into a huge, say a quarter the size of the universe, black hole with a bridge to another region? I don't know, the CMB measurements may put a limit on how much of that is possible, seems to me it would show some huge CMB variance, so the question would be what limits can one place on those.  
A: There are two ways in which to interpret "shape". The first is topological shape - global connectivity (hence the joke that a topologist can't tell a cup with a handle from a donut with a hole) and the other is geometric shape - local sizes and angles (an espresso cup and a teacup have different geometric shapes, but the same topological shapes).
If a wormhole could be created, it would change the topological shape of the universe (and induce its own geometry as well, of course) and it is a famous result of Robert Geroch [1] that any topology change necessarily leads to the formation of Closed Timelike Curves (~"time machines"), the possibility of which led Hawking [2] to propose the Chronology Protection Conjecture to protect the universe against CTCs due to wormholes... i.e. wormholes are banned because they would change the topology and so create time machines, which are awkward for physics.
Thus the answer to your question is: if the Chronology Protection Conjecture has force, then wormholes cannot be created or destroyed because that would change the topological shape of the universe and so violate chronology. However, "eternal" wormholes, i.e. constant topology, are permissible [neglecting other arguments about creating time machines from wormhole through other relativistic effects.]
[1] R. P. Geroch, “Topology in General Relativity,” Journal of Mathematical Physics, vol. 8, no. 4, p. 782, 1967.
[2] S. W. Hawking, “Chronology protection conjecture,” Phys. Rev. D, vol. 46, no. 2, pp. 603–611, Jul. 1992.
