Yes, you can install UpdraftPlus on your clients’ sites. There are no technical or licensing restrictions preventing this – we are all in favour of it.
Since you are our customer (and your client is your customer), if there are support issues that come, then it is you who has the right to come to us for personal support. Your client, if they need to, will go to you for support. Of course, it is also very acceptable for your client to get their own UpdraftPlus licence so that they can get direct support, if you prefer that. Whatever works best for you is fine with us!