Yes, web-based Unity content can save data to the local machine using the PlayerPrefs class. Each webplayer url can save up 1 megabyte of data, consisting of either ints, floats or strings. For example:

// to save an integer value
PlayerPrefs.SetInt("Player Score", 10);

// to retrieve the integer value
print (PlayerPrefs.GetInt("Player Score"));

And the commands are very similar for floats and strings:

PlayerPrefs.SetFloat("Player Lap Time", 29.3);
print (PlayerPrefs.GetFloat("Player Lap Time"));

PlayerPrefs.SetString("Player Name", "Duck");   
print ("Welcome back, " + PlayerPrefs.GetString("Player Name") + "!");

You can also query the player prefs data to see whether a certain key has been set, using PlayerPrefs.HasKey, and delete keys using either PlayerPrefs.DeleteKey or PlayerPrefs.DeleteAll.