Update: There is now an official API: https://developer-programs.linkedin.com/documents/company-search
The official API does not seem to have this feature. However they do have a URL where you can pass in a search for companies and it will return a JSON response with the company name, logo URL, number of employees, and a link to the company profile.
For instance a search of "citr" would be
http://www.linkedin.com/ta/federator?query=citr&types=company,group,sitefeature
Which returns
{
"sitefeature":{
"resultList":[
]
},
"company":{
"resultList":[
{
"id":"1195260",
"headLine":"<strong>CiTR</strong> SHINDIG",
"displayName":"CiTR SHINDIG",
"subLine":"Entertainment; 1-10 employees",
"url":"http://www.linkedin.com/companies/1195260"
},
{
"id":"2641",
"headLine":"<strong>Citr</strong>ix Systems",
"imageUrl":"http://media03.linkedin.com/mpr/mpr/shrink_40_40/p/3/000/043/32b/0076ade.png",
"displayName":"Citrix Systems",
"subLine":"Computer Software; 1001-5000 employees",
"url":"http://www.linkedin.com/companies/2641"
},
{
"id":"7659",
"headLine":"<strong>Citr</strong>oen",
"imageUrl":"http://media01.linkedin.com/mpr/mpr/shrink_40_40/p/3/000/01d/3d0/277c145.png",
"displayName":"Citroen",
"subLine":"Automotive; 10,001+ employees",
"url":"http://www.linkedin.com/companies/7659"
},
{
"id":"260214",
"headLine":"PSA Peugeot <strong>Citr</strong>o?n",
"imageUrl":"http://media02.linkedin.com/mpr/mpr/shrink_40_40/p/2/000/021/0b7/07c4193.png",
"displayName":"PSA Peugeot Citro?n",
"subLine":"Automotive; 10,001+ employees",
"url":"http://www.linkedin.com/companies/260214"
},
...
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