Yes, you should definitely tell your manager about your ambitions.
So ask her for a 1:1 meeting. Tell her you would like to talk about your career development.
During that meeting, tell her that you are happy with your current position, but you think you might be able to do even more for the company. So for the benefit of the company would like to develop yourself into a tech lead position. Ask her what she thinks you need to do to qualify yourself for such a position.
She might either be supportive and appreciate your initiative. Or she might try to brake you down with platitudes about how you are doing such a good job in your current role. But in any case, she will be unable to make you any promises on the spot. After all, she will have to consider other business interests first and/or consult the managers above her.
So before you leave the room, ask for a followup meeting to discuss a detailed career plan with a road map and milestones. You might want to write your own proposal for such a plan and then use the meeting to negotiate an agreement between her proposal and yours. This proves that you are serious and that you have put thought into how to advance your career.
Just keep in mind that there will never be a guarantee that you will ever get that promotion. There might not be an open position. Or when there is an open position, there might be someone who they consider an even better fit than you. So always stay on the lookout for better career opportunities in other companies.