They have long tap roots and if they get damaged they will not survive. You will have to dig down a long way if it is a well established aquilegia. Best time to do it is when they are small I think. Someone else will maybe give you some better advice. Otherwise save some seed, they do not always come true to colour and form when you get them to germinate but it is well worth a try. Good luck with it. :O)

I don't think I've ever killed an aquilegia by moving it, even if I only get 3 inches of taproot, so my advice would be move at any time as long as you can be sure to pay attention to watering for the next 3 months - ie leave until autumn if you will be away for 10 days in August.
If you do damage the taproot, there are also fibrous roots to assist, so as long as you take off any flower stalks and maybe a third of the leaves I think it'll be fine!

I moved a one year old one in the autumn last year very successfully.

I moved a Yellow Aquiligia last year, it was quite tall, and It has just started to flower and yes true to form yellow.