Just a refinement on David's answer really, but this is more precisely what it means.

Supply chains means the sources of the goods and services they obtain, not only the immediate supplier but also those who supply the supplier:

"A supply chain is a system of organizations, people, activities, information, and resources involved in moving a product or service from supplier to customer. Supply chain activities transform natural resources, raw materials, and components into a finished product that is delivered to the end customer. "
http://en.wikipedia.org/wiki/Supply_chain

So it means any person or entity involved at any stage in producing whatever is supplied to the faiths (religious institutions) in question.

Modern slavery is a compound expression denoting kinds of practice regarded as slavery according to modern definitions, which are somewhat broader than traditional slavery on the model of the historical slave trade. The term covers bonded labour, child slavery, early and forced marriage, forced labour, descent-based slavery and trafficking. See here for further details:
http://www.antislavery.org/english/slavery_today/what_is_mod...

"Modern slavery-proof" is constructed from "modern slavery" and "proof", by analogy with "waterproof". It something is waterproof it is inaccessible or impervious to water, protected from water. Water cannot enter it. It can be an adjective (a waterproof coat) or a verb (to waterproof a coat).

So by analogy, to "modern slavery-proof" (verb) a supply chain means to make it impervious to modern slavery, in the sense of ensuring that modern slavery does not enter it, is not part of it, at any point. If this is done, the supply chain is "modern slavery-proof" (adjective): not only does it not involve any modern slavery, but steps have been taken to ensure that it doesn't. In your context it is a verb.

So "modern slavery-proof their supply chains" means take steps to ensure that they do not obtain any goods or services in whose production or distribution slave labour has been used at any stage, as defined according to the concept of modern slavery.

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Note added at 4 hrs (2014-03-13 05:06:35 GMT)
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"all global faiths to modern slavery-proof their supply chains and investments" is an elliptical way of saying "all global faiths ARE to modern slavery-proof their supply chains and investments", in other words they must do so.

Note that it also refers to investment: faiths must not invest in any company that is directly or indirectly involved with modern slavery, either using slave labour themselves or working with other companies that do.