Can I avoid downtime while transferring my existing hosting account to you?

Yes, you can. To avoid downtime while transferring your hosting account to us you will need to do the following:

First step will be ordering one of our hosting packages which suits your requirement.
Do not cancel your previous hosting account!
Once your package is active and Welcome email is received, upload your content to our server.
Once the content is uploaded, re-point nameservers to the ones indicated in your Welcome email.
As a result your domain will be resolved to your previous host from the places where the propagation has not been completed yet and to a new one – from the places where it has already passed.
Once the propagation is finished in all over the world you may delete the content on your old hosting.

You can find additional information on how to transfer your hosting account to us here.