can flask use with jquery post?

$.post() doesn't use JSON. It sends a regular POST request, with application/x-www-form-urlencoded as the content type. In a Flask route this means that the fields you posted are found in the request.form object. If your view returns HTML you can insert it into your HTML document.

Just pick a route name that reflects what you want to do:

@app.route('/turn_servo', methods=['POST'])
def turn_servo_ajax():
    direction = request.form['direction']

    # do something with the direction value; it is a string
    if direction == 'left':
        # ...
    else:
        # ...

    return '<div>Turned the servo 50 degrees to the {}</div>'.format(direction)

You then use that route name in your $.post:

$('#left_button').click(function(){
            $.post("/turn_servo", {direction:"left"}).done(function (reply) {
                $('#camerapos').empty().append(reply);
                alert("left button clicked");});

        });

I've copied the code from your original question to demonstrate how the above answer (Martijns) works with your code.

# Save this file as app.py
from flask import Flask, jsonify, render_template, request
from cameraservo2 import your_cam_function

app = Flask(__name__)

@app.route('/turn_servo', methods=['POST'])
def turn_servo_ajax():
    direction = request.form['direction']
    cam_result = your_cam_function(direction=direction)
    return '<div> {} </div>'.format(cam_result)   

if __name__ == '__main__':
    app.run(debug=True)


# This code is in cameraservo2.py and is imported above.
# I've simplified it to a function (not a member of a class).
# You have to work out how to use it in your context.
def your_cam_function(**data):
    import pigpio
    import time

    servos=4
    key = data['direction']

    m=1500
    while (m >= 500 and m <= 2500):
        if (key =="left"):
            m=m+100
        elif (key =="right"):
            m=m-100

    pigpio.start()

    pigpio.set_servo_pulsewidth(servos, m) 
    servostatus= "Servo {} {} micro pulses".format(servos[0], key, m)
    print servostatus
    time.sleep(1)

    pigpio.stop()

    return servostatus

Now when you do a POST request to /turn_servo and give it a direction: left, it will call the function from your cameraservo2.py. The turn_servo_ajax-function will return a piece of HTML to the user, <div>Servo x y micro pulses</div>.