“Avec un peu de chance” is to me most idiomatic way to translate “hopefully”.

I never saw “avec espoir” used actually, and well even if it doesn’t seems like it is, “avec un peu de chance” is often used, and perfectly accurate.

I think @servabat is right. Another way would be to reformulate the whole sentence.

• On peut espérer que…
• En espérant que…
• J’ai bon espoir de…
• Il y a fort à parier que…

Probably less idiomatic than the first two good answers and their comments, but I often use the imperative/suggestive form “Espérons que” to express the notion of “hopefully”/(“Let’s hope that …”):

“Espérons que tout ira/aille bien”

(But if you’re all alone talking or thinking to yourself [and you don’t have a tapeworm], I suppose this plural notion might be out of place and/or taken as an example of the “royal we”).

i’m french and i agree with “espérons que” (let’s hope that…). – sorry i can’t comment nor vote Papa Poule’s answer up yet because i don’t have enough reputation.

There is no word to translate “hopefully” literally and i think you should avoid to stick to find one, but construct the sentence differently (with “espérons que”)

Sometimes something as simple as:

j’espère que, on espère que

You can complement the sentence (without “hopefully”) with “croisons les doigts”, or “espérons-le”/”Je l’espère”, or “soyons optimiste”. For instance, with

Hopefully, this answer will get some votes:

• Soyons optimistes, cette réponse recevra quelques votes,
• Cette réponse recevra quelques votes, je l’espère,
• J’espère que cette réponse recevra quelques votes,
• Croisons les doigts, cette réponse recevra quelques votes.