'He can score' here means 'he can persuade a woman to have sex with him'. The (fairly big) clues are in the text preceding and following that which you posted.
Gene Simms was working nights at the box factory, and the two began passing afternoons together. Driving his car or riding in Ernie’s, haggard, frowning, yawning, smoking with yellowed unsteady fingers, a blond oily lock hanging over his forehead, Gene talked mostly on the same subject, his descriptive powers arousing in Ernie a curious agitation and a fear that what he had with Faye might be of a quality below the possible or even the usual. “Well, I don’t know.”
“But you got to know what you’re doing.”
“That’s right, sure, they won’t go for just anybody.”
“If the right guy comes along he can score.”
“There isn’t a one that don’t want it,” said Gene.
“Everybody’s got a mate somewhere.”
“I don’t care who it is. You know Eleanor MacDonald? I plugged her.”
“I know, you told me.”
“You got to understand their minds. If you can get your knee between their legs you’re usually on your way.”
From 'Fat City' by Leonard Gardner
The subject is sex, and how to persuade women to let a man have it with them (success at this is one meaning of 'scoring'). 'Plugging' is a crude and vulgar way of referring to the sex act, and this points to the type of person Gene is, and his attitude towards women. This and the language that Gardner uses when describing him ('oily lock', 'yellowed unsteady finger') paints a picture of a sordid and unsavoury character.
Score (verb) [ Intransitive ] slang
to have sex with someone that you have usually just met:
Did you score last night, then?