The "best" option is a matter of perspective. 

The BOI is open (but limited to a machine on the same local network as the Sage server, which you probably don't want to expose to the internet as Sage100User indicates), with a high learning curve, and using someone else's expertise will save you a lot of headaches. 

Sage's eBWS is very limited in functionality, but we did implement this once.

CIM Cloud / WSP is a high quality full service solution.

Insynch is a bridge we've used for a couple clients, and I understand MBA has a web API too.

Sage is also working on an Office 365 connector if you want to look into that.  (The demos look great).

Plus there is the file exchange option, which is probably our most common method of working with web orders.  Importing from orders downloaded as CSV (or data pushed into a staging SQL table), with read-only ODBC to provide data.

The links to find this stuff is

CIM Cloud's API (Most use their eCommerce site but this link is about their API to the Hub) Data Hubs & API | cimcloud.com

 Sage Data Exchange - by MBA is a great way to get JSON or XML from Sage 100 to integrate to other systems.

InSync is more of a Sync tool but they are used a lot for eCommerce sites Integrate Your Website (roi-consulting.com)

The Office 365 is really cool but is limited. They are developing for it so I would expect it to continue to expand as time goes on. It shows very well. It does require a dive into the Microsoft eco system of online apps.

Sage eBusiness Web Services is a module that can provide some limited access through an API. I do not believe they are doing any development work on it. No changes are expected.  All for Sage e-Business Manager - e-Business Web Services (4ebm.com). This is part of Sages eBusiness Module which is a built in eCommerce Site. (Great, out of the box, module to get up and running with eCommerce) eBusiness hasn't been changed in many years and the eCommerce has a late 1990's look to it. It can be customized and developers do make changes to it.

Using BOI limits you to be on the local machine but does give you a lot of access to what you need. You would have to have some kind of wrapper or something to provide access for external access.

Sage also has SDATA for Sage 100. It is not really being developed and I think is still in version 1 for Sage 100.

Visual Integrator can be used for Import/export of Files.

ODBC can be used to query (read only) data from Sage 100

Each option has it positive and negative sides. Some are built in (BOI, SData, ODBC), some are additional modules ( and others are 3rd party. Some of the options may also have an annual maintenance or subscription cost to it. Some will require more of an understanding of Sage 100 to make it all work (BOI, ODBC, Visual Integrator)

The links to find this stuff is

CIM Cloud's API (Most use their eCommerce site but this link is about their API to the Hub) Data Hubs & API | cimcloud.com

 Sage Data Exchange - by MBA is a great way to get JSON or XML from Sage 100 to integrate to other systems.

InSync is more of a Sync tool but they are used a lot for eCommerce sites Integrate Your Website (roi-consulting.com)

The Office 365 is really cool but is limited. They are developing for it so I would expect it to continue to expand as time goes on. It shows very well. It does require a dive into the Microsoft eco system of online apps.

Sage eBusiness Web Services is a module that can provide some limited access through an API. I do not believe they are doing any development work on it. No changes are expected.  All for Sage e-Business Manager - e-Business Web Services (4ebm.com). This is part of Sages eBusiness Module which is a built in eCommerce Site. (Great, out of the box, module to get up and running with eCommerce) eBusiness hasn't been changed in many years and the eCommerce has a late 1990's look to it. It can be customized and developers do make changes to it.

Using BOI limits you to be on the local machine but does give you a lot of access to what you need. You would have to have some kind of wrapper or something to provide access for external access.

Sage also has SDATA for Sage 100. It is not really being developed and I think is still in version 1 for Sage 100.

Visual Integrator can be used for Import/export of Files.

ODBC can be used to query (read only) data from Sage 100

Each option has it positive and negative sides. Some are built in (BOI, SData, ODBC), some are additional modules ( and others are 3rd party. Some of the options may also have an annual maintenance or subscription cost to it. Some will require more of an understanding of Sage 100 to make it all work (BOI, ODBC, Visual Integrator)