When the family FIRST receives news of the elopement, they are horrified. So I think it is the elopement itself that is dreadful--of course, later it proves to be even worse than they first thought--they discover that the marriage may not have even taken place. But the INITIAL horror is strictly over the elopement.

If a woman marries without parental consent, she has usurped the authority of her father, who has the legal obligation to make the decision for her and to see that her interests are protected.
In that society, she would be seen as bold and brazen to step outside of her father's authority, and these qualities were not prized in that time! So, she has shamed herself and her father.
If she had continued on to Gretna Green and entered into a marriage, she would have stripped herself of legal rights in the marriage settlement...obviously, her father has little bargaining power for Lydia's financial interests if she is already married...or if she has compromised her virtue. And of course, if the scoundrel has ulterior motives and doesn't do the honorable thing...well, Lydia is now unmarriable.

In any event, no matter HOW the thing had turned out, Lydia would not be able to marry any man of honor.

So the elopement is horrifying because she has put her future into the hands of a man who doesn't care at all for her reputation or honor (or he wouldn't have agreed to elope with her!)...and because she has publicly dishonored her father.