I would recommend using parsers for parsing and amending dynamic JSON data. However, if the JSON is static and you want to achieve your result using regex; then your regex is very correct(although it can be compacted. See in answer).  You just need to use LocalDateTime instance instead of LocalDate instance. Something like;
// Sample code. Please manipulate according to your requirements.
import java.time.*;
public class Main
{
    public static void main(String[] args) {
        // Sample test String
        String content = "\"buyDate\":\"2019-04-09T13:21:00.866Z\"";
       // String result = content.replaceAll("(\"buyDate\":\"(\\d{4}-\\d{2}-\\d{2}.*)\")", "\"buyDate\":\"" 
        //+ LocalDateTime.now().toString() + "Z\"");
// COMPACTED REGEX
        String result = content.replaceAll("(\"buyDate\":\".*?\"", "\"buyDate\":\"" 
        + LocalDateTime.now().toString() + "Z\"");
        System.out.println(result);
    }
}
// Outputs: "buyDate":"2020-06-16T09:33:38.276Z" a.t. online IDE's local time.
You can find the sample run of the above code in here.
If you want to replace only date value leaving time and zero-hour format as it is(although I don't see any practical reason to do so); then you could try this:
import java.time.*;
import java.time.format.*;
import java.util.regex.Pattern;
import java.util.regex.Matcher;
public class Main
{
    public static void main(String[] args) {
        LocalDate dt = LocalDate.now();
        final String regex = "\"buyDate\":\"(\\d{4}-\\d{2}-\\d{2})T";
        final String string = "\"buyDate\":\"2019-04-09T13:21:00.866Z\"";
        DateTimeFormatter formatDateToday=DateTimeFormatter.ofPattern("YYYY-MM-dd");
        final String subst = "\"buyDate\":\""+ dt.format(formatDateToday) +"T";
        final Pattern pattern = Pattern.compile(regex, Pattern.MULTILINE);
        final Matcher matcher = pattern.matcher(string);
        // The substituted value will be contained in the result variable
        final String result = matcher.replaceAll(subst);
        System.out.println(result);
    }
}
- I used DateTimeFormatterclass for formatting the Date.
- I used PatternandMatcherclass to match only the dates in the given JSON string.
You can find the sample run of the above code in here.