My HB level is currently 10.5-. As a female vegetarian who does endurance mountain biking I battle to keep iron levels constant. May I keep training?

Absolutely, but.... Exercise induced anemia is part of the body's normal adaptation to endurance exercise and is expected. The challenge is to make certain that you do not also have iron deficiency anemia or some other cause. One way to tell is by showing that your HB level moves towards normal after a period of limited exercise, say a week or more. If concerned see Dr.
Yes. You can train as you've likely conditioned your body to release oxygen from those red cells more efficiently and conditioned your end organs to use the oxygen more completely. That being aid, vegetarians often have trouble with iron and B12, both of which are important for RBC production.
Take iron. It Is likely that you are iron deficient and need to take oral iron supplements on a regular basis. You may keep training and training will be more effective if your hemoglobin is normal. Taking iron supplements will not negate any of the benefits of a vegetarian diet.