How to make certain mathematical characters be written as bold by default?

Yes, you can use InputAliases to style the replacements as you wish. I explained how to make auto replacements in this answer, but that was for a more specialized purpose.

As a simple example, you can do something like:

SetOptions[SelectedNotebook[], InputAliases -> 
    {"foo" -> MakeBoxes@Row[{
        Style[foo, FontColor -> Red, FontWeight -> Bold, 
            FontSlant -> Italic, FontFamily -> "Verdana"],
            \[VeryThinSpace]
        }]
    }
]

Now if you type EscfooEsc, it will show up as a bold, italic red foo.

Perhaps you can use the following alias instead:

CurrentValue[EvaluationNotebook[], {InputAliases, "vec"}] = OverscriptBox[
    StyleBox["\[SelectionPlaceholder]", FontWeight->Bold],
    "\[RightVector]"
];

Then, using the following key strokes: Esc v e c Esc Tab b produces:

where the letter b is bolded.