Unfortunately, it is all in Hebrew, so far.
If you want to follow it anyway, the Twitter account is @IDF1967.
If you want to complain to the IDF to have it translated into English, just retweet this.
Alternatively, if you want to spend the next week translating the hundreds of tweets yourself in near real-time and you want to create an English-language Twitter account for that purpose, let me know and I'll publicize that. You can call it Naksa1967 or something like that. It can have the added bonus of upsetting people who are sorry that Israel won.
הסירנות עלולות להישמע בכל רגע #יוני67 vine.co/v/b3vPBaqWruv
— צה''ל בששת הימים (@IDF1967) June 3, 2013
(h/t Gidon Shaviv)
UPDATE: Google Chrome users with the translate plug-in can see a reasonable translation in real time on the page. (h/t Clark)