How to encode a text input for a URL?

Why not pass it in the query parameters directly instead of altering the URL on run time? 
Query parameters are designed for this purpose only. 

you're not saying exactly how you're using the text but on the assumption that you do need to parse the value and replace a set of special characters with some corresponding values here's an example of how you might go about it:

a!localVariables(
  /* define a list of the special characters that needs replacing */
  local!specialChars: {
    " ",
    char(34),/* unicode for quotation mark */
    char(39) /* unicode for apostrophe */
  },
  /* define a list of the replacement characters - must be in the same order */
  /* as thelist of special characters */
  local!replaceWithChars: {
    "%20",
    """,
    "'"
  },
  /* use fn!reduce to pass the result of eahc iteration into the next iteration */
  fn!reduce(
    fn!substitute,/* use substitiue to replace the special characters */
    ri!myString,
    fn!merge( /* use fn!merge to create a "list of lists" */
      local!specialChars,
      local!replaceWithChars
    )
  )
)

Note:

Line 12 should read """ but is obviously being parsed by this web-based app to read """

Line 13 should read "'" for the same reason