I've seen the reading of 株式会社 written both with rendaku and without. I can't tell whether the version without rendaku is simply a typo, or a true variant of the word.
Although かぶしきかいしゃ is not wrong, かぶしきがいしゃ is more common in recent years, as explained in other answers.
When the word 会社 follows a noun and forms a compound noun, it's usually read がいしゃ. For example,
大会社 だいがいしゃ etc.
However, there are a few exceptions.
物的会社 ぶってきかいしゃ etc.
There are entries from 9 dictionaries on kotobank.jp. Three give no explicit reading, three say かぶしきがいしゃ, and three say かぶしきかいしゃ. Could it be any more inconclusive?
I think I always hear it with rendaku, and two of the nine dictionary entries are from 大辞林 and 大辞泉, two popular monolingual dictionaries. These two may not be entirely independent indicators, but I would go with かぶしきがいしゃ, with rendaku. Still, I wouldn't go as far as calling かぶしきかいしゃ a typo.
(As with all "reading" questions, it seems particularly hard to find any corpus data, because the reading is never spelled out. Cf. 今日 こんじつ こんにち きょう or 私 わたくし わたし etc.)
株式会社 is pronounced both kabushiki kaisha and kabushiki gaisha. But it is predominantly pronounced kabushiki gaisha today in Japan. By the same token, more people call 投資信託会社 (investment trust company) tōshi shintaku gaisha instead of tōshi shintaku kaisha.
When you voice 株式 and 会社 separately, you should pronounce them kabushiki and kaisha individually.
We call the sounds of か行（syllabic line）— か、き、く、け、こ (ka, ki, ku, ke, ko) — 清音 (literally "clear sounds"), and its voiced (vowel and consonant) line — が、ぎ、ぐ、げ、ご (ga, gi, gu, ge, go) — 濁音 (literally "unclear sounds"). We don't use 濁音 for 株式 on its own.
Although this dictionary says 株式会社 as カブシキ・カイシャ，I have never ever met anyone said in my enrire career sombody in a company said "Our company's name is カブシキ・カイシャ XXXXXX，or XXXXX カブシキ・カイシャ" ( Sometimes, in business mannars, whether the name of the company should be placed before カブシキ・ガイシャ or after do matter. ( For example, when you are trying to send invoices to them. ) And it is really hard to say, ( as an example, カブシキ・カイシャ・トヨタジドウシャ ( People of Toyota could think of you a strange guy, in my personal guess unless you are not a native Japanese. ))
It is commonly held in the business world, I am enough certain that people would not call a company XXXXXX カブシキ・カイシャ or カブシキ・カイシャ XXXXXX.