I am not quite sure what do you mean by "update the string variable without swiping another card"
A$ = INPUT$(1) lets the program read the CR-terminated string in the serial buffer. Presumably when you swipe the card the card reader sent the CR-terminated string to the PLC's COMM1 buffer. A$ = INPUT$(1) is then used to read the string.
Once it is read the string is removed from the buffer but will stay in A$. You can then display the string using SETLCD on the LCD and you can process the data in the A$ e.g. using MID$, VAL, HEXVAL etc to convert to integers.
A$ data will not change unless you change it by re-assigning it. If you continuously execute A$ = INPUT$(1) and there is nothing in the serial buffer then A$ will get a blank string. So the way to write the program is to read the data into a temporary string and then if there is data, store it into another string:
E.g.
B$ = INPUT$(1)
IF LEN(B$) <> 0
A$ = B$
ELSE
RETURN
ENDIF
SETLCD 1,1, A$