R, 77 bytes

f <-function(c)sum(unlist(gregexpr(c,capture.output(dump("f","")),f=T))>0)/77

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Uses R dump(,"") function to output own source code*, capturing output (normally to the console) using capture.output(). Then finds the indices of all matching characters with gregexpr(,fixed=TRUE) ('fixed' option [abbreviated in golf code to 'f'] prevents characters such as '(' from being interpreted as part of a regular-expression), and report the sum() of matching positions >0 (non-matching positions are output as -1), divided by the source code length (77 bytes).

*Note that the TIO interface gives slightly differently-formatted output from the dump() function to running directly in R, so counts of whitespace characters are inaccurate in TIO (but correct in R).

R (without reading own source code), 106 bytes

'->p;sum(gregexpr(scan(,""),sQuote(p),f=T)[[1]]>0)/53'->p;sum(gregexpr(scan(,""),sQuote(p),f=T)[[1]]>0)/53

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Same approach as above, but including quoted string instead of reading the source code itself.

R, 77 75 bytes

Edit: -2 bytes thanks to Giuseppe

f <-function(c)sum(c==unlist(strsplit(capture.output(dump("f","")),"")))/75

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Uses R dump(,"") function to output own source code*, capturing output (normally to the console) using capture.output(). Then finds the number of matches to the function argument c, divided by the source code length (75 bytes).

*Note that the TIO interface gives slightly differently-formatted output from the dump() function to running directly in R, so counts of whitespace characters are inaccurate in TIO (but correct in R).

R (without reading own source code), 106 bytes

'->p;sum(gregexpr(scan(,""),sQuote(p),f=T)[[1]]>0)/53'->p;sum(gregexpr(scan(,""),sQuote(p),f=T)[[1]]>0)/53

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Same Similar approach to above, but including quoted string instead of reading the source code itself.

R, 77 bytes

f <-function(c)sum(unlist(gregexpr(c,capture.output(dump("f","")),f=T))>0)/77

Try it online!

Uses R dump(,"") function to output own source code*, capturing output (normally to the console) using capture.output(). Then finds the indices of all matching characters with gregexpr(,fixed=TRUE) ('fixed' option [abbreviated in golf code to 'f'] prevents characters such as '(' from being interpreted as part of a regular-expression), and report the sum() of matching positions >0 (non-matching positions are output as -1), divided by the source code length (77 bytes).

*Note that the TIO interface gives slightly differently-formatted output from the dump() function to running directly in R, so counts of whitespace characters are inaccurate in TIO (but correct in R).

R, 77 bytes

f <-function(c)sum(unlist(gregexpr(c,capture.output(dump("f","")),f=T))>0)/77

Try it online!

Uses R dump(,"") function to output own source code*, capturing output (normally to the console) using capture.output(). Then finds the indices of all matching characters with gregexpr(,fixed=TRUE) ('fixed' option [abbreviated in golf code to 'f'] prevents characters such as '(' from being interpreted as part of a regular-expression), and report the sum() of matching positions >0 (non-matching positions are output as -1), divided by the source code length (77 bytes).

*Note that the TIO interface gives slightly differently-formatted output from the dump() function to running directly in R, so counts of whitespace characters are inaccurate in TIO (but correct in R).

R (without reading own source code), 106 bytes

'->p;sum(gregexpr(scan(,""),sQuote(p),f=T)[[1]]>0)/53'->p;sum(gregexpr(scan(,""),sQuote(p),f=T)[[1]]>0)/53

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Same approach as above, but including quoted string instead of reading the source code itself.

R, 77 bytes

f <-function(c)sum(unlist(gregexpr(c,capture.output(dump("f","")),f=T))>0)/77

Try it online!

Uses R dump() function to output own source code*, capturing output (normally to the console) using capture.output(). Then find the indices of all matching characters with gregexpr(,fixed=TRUE) ('fixed' option prevents characters such as '(' from being interpreted as part of a regular-expression), and report the sum() of matching positions >0 (non-matching positions are output as -1), divided by the source code length (77 bytes).

*Note that the TIO interface gives slightly differently-formatted output from the dump() function to running directly in R, so counts of whitespace characters are inaccurate in TIO (but correct in R).

R, 77 bytes

f <-function(c)sum(unlist(gregexpr(c,capture.output(dump("f","")),f=T))>0)/77

Try it online!

Uses R dump(,"") function to output own source code*, capturing output (normally to the console) using capture.output(). Then finds the indices of all matching characters with gregexpr(,fixed=TRUE) ('fixed' option [abbreviated in golf code to 'f'] prevents characters such as '(' from being interpreted as part of a regular-expression), and report the sum() of matching positions >0 (non-matching positions are output as -1), divided by the source code length (77 bytes).

*Note that the TIO interface gives slightly differently-formatted output from the dump() function to running directly in R, so counts of whitespace characters are inaccurate in TIO (but correct in R).