do native English speakers completely omit the /r/ sound in /ˈev·riˌwʌn/ (everyone)?

No. Just... no. The only people I've heard omitting the r sound in everyone either have a speech impediment, or are affecting one. I can't even think of a regional accent where that would be the case, though perhaps the deep southern USA may, possibly, pronounce it less clearly than other parts of the English speaking world. (It may sometimes sound like "eveh-won", although even then most southern speakers I've heard would be closer to "everh-won".)

I don't know which dictionary this is, but either you've misinterpreted it, or I think you need a different dictionary.

Here in Texas, I think "everyone" is pronounced "ya'll"... :)

Seriously though, the 'r' should be there if you listen for it.