Opening the source code of such a website or providing reliable and trusted audits to guarantee the proper function (I consider the latter insufficient, but customers will decide…) is up to the owner of the website.
I’m not aware of such a website being part of the development road map of BCE (albeit the design paper might lead to the conclusion that BCE will provide such a website “A web based interface that will be familiar to users of centralized cryptoasset exchanges will be offered.”, page 1, second paragraph).
As there’s no development road map that comes as no surprise that I’m not aware of it
I expect third party providers to provide this service (likely for a fee).
What BCE is going to provide is a client that does the job. This client will be used by website providers.
Foresight:
tech savvy people will want to create their own website to access the blockchain via web interface (e.g. from smartphone). The challenge doing that shouldn’t be very big.
Foremost you need a web site that controls the BCE client via RPC. That’s most of it.
Things like “owncloud” plugins or alike might be created to make this convenient.
Or “RaspBCE” for RaspberryPi. Download image, dd it to the RaPi, power up the RaPi, forward TCP/443 (or whatever) to the RaPi.
Your own private BCE web site is running!
Share it with your friends, install a paywall (require x NBT to be sent an address to receive an access code via OP_RETURN message) for paid access or allow free access for all the world.
DigitalOcean droplets (just a random example for virtual servers), installation packages for Linux distributions, installer for other operating systems, etc. are possible as well.
Via BCE exchange web sites you effectively can create something similar to e.g. blockchain.info, but in an improved version:
Web access to BCE provides BCE with a kind of online wallet capabilities - similar to blockchain.info.
Transactions (e.g. withdraw) can be executed using a web site - similar to blockchain.info.
This requires coins to be available in a multi signature deposit address - assumably better security that at blockchain.info (does blockchain.info support multi signature addresses and if what level?).
You can execute payments with all coins supported by BCE by creating a withdraw (to the payment deposit address) on any BCE exchange web site - that’s way more flexible than at blockchain.info which only supports BTC.
And if you don’t have enough BTC, trade some of your deposited LTC or whatever first - that’s way more flexible than at blockchain.info.
But the major benefit over central wallets is this:
there’s no risk to get your funds seized by taking down a central service or stolen by that central service.
The central server as single point of failure is gone.
All relies on the (distributed) reputed signers which can’t be tracked easily if ever.
If you host your own web site, this is a quite secure way to create this kind of online wallet.
For all who are already ensnared by the beauty of NBT, NuDroid is the more elegant way to do that.
Legions of people who use blockchain.info or similar will find an alternative in BCE.
It will be necessary to verify the integrity of all those ways to create a BCE web interface with a few clicks or command lines.
Educated BCE users should be aware of associated risks and use the BCE native client when handling big amounts,
Convenient ways to access BCE (via web access) will be created if BCE is successful!
Much of the above is dreaming, but I bet some of it will come true.