"Undue influence" is a legal term used when someone is taking an advantage over another person using his social or professional position. (See http://en.wikipedia.org/wiki/Undue_influence.) For example, I might have to reveal something I do not usually tell anyone to my doctor. If he used such information against me, he would commit an "undue influence".
"Vague charges" means the charges are not concrete enough. The word vague means uncertain, or unclear.
The sentence "He shuffles out of the consequences by vague charges of undue influence" might look a bit unfamiliar (OAD noted the definition as archaic). It also looks almost like it needs a verb-ing after that by, e.g. "avoid the consequences by denying...", "avoid the consequences by admitting...", "avoid the consequences by having the court hold the claims...". How should we understand the sentence, which has no verb-ing after by?
@WendiKidd's suggestion clears this up. You should interpret the sentence as: he can avoid the consequences by claiming that someone else unduly influenced him (and made him do the actions he should be in trouble for).
In case you get confused (as I did) and wonder why "shuffles out of ... by ..." should be interpreted that way, please read on.
To gain the insight for such usage, besides recalling some simple phrases such as "he goes to school by bus," I searched the web and found some interesting similar uses. Having full passages, the contexts should be much clearer. Here are some of them:
if it has managed to avoid the consequences by way of vertical integration, then I am slightly jealous. (ref)
So if we want to foster innovation in students we need to learn to accept massive, continual failure. Else, we simply will avoid the consequences by inertia. If you do nothing you run no risk of failing in doing something. (ref)
[excerpt from a chat log] (ref)
How about we just stop the whole ridiculous idea of DST!?
Damn that's tonight. I hate this time change.
This is the body's natural time. It's Daylight Savings Time that is the killer.
I know how we can avoid the consequences by no more day light saving time!