No, if you can’t enter double quotes you cannot escape from the attribute value. See the WHATWG spec for details on how browsers parse attribute values. The possible parser state transitions in that state are:
||end attribute value|
||parse a character reference|
|NULL||error, but will continue parse|
|EOF||error, will end parse|
|other characters||added to attribute value|
So unless there is a bug in the HTML parser, you cannot escape. The
& ampersand is the only interesting character in this context, but it does not lead to parser states that would terminate the attribute value. A null character will be substituted with the U+FFFD replacement character, so it will not lead to interesting behaviour in the browser.
However, the web application will do something interesting with that input value. Maybe this “something” will have exploitable vulnerabilities if you can provide a specially prepared value.