The most common and most likely answer to this crossword clue is the 6 plus 7 letter word SILVER SURFERS
I've seen this crossword clue in The Times
S
I
L
V
E
R
S
U
R
F
E
R
S
More clues leading to the result SILVER SURFERS
Buddy explains!
The straight part
We know that "They are using the Internet and getting on with it" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case they are using the internet and getting on with it = silver surfers.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.