Presumably, as a Japanese name, it should be written in Chinese characters or in hiragana.
It's a matter of stylistic choice which is hard to explain logically. It's kind of like asking why many Western companies adopted lowercase logos these days, which apparently look "grammatically wrong" to English learners :-)
In general, katakana names often have a "international", "modern", or "technical" impression, which is definitely good for a high-tech company like Toyota. On the other hand, kanji names tend to look somewhat "old", "classic" or "traditional" to the eyes of Japanese people. If you have the textbook knowledge of when to use katakana in everyday writings, you can probably feel why it's so.
As a matter of fact, a majority of technology companies prefer katakana or alphabet branding over kanji branding even when their official names are in kanji. Toyota and Mazda switched their official names to katakana many years ago. Honda retains the official company name in kanji, but we rarely see the kanji name in news and ads. Some companies like Nissan still use kanji along with alphabet.
To take another example, beer/whiskey breweries in Japan tend to have katakana names/brand, (キリンビール, アサヒビール, サッポロビール, ...) while sake/shochu makers tend to have kanji names (白鶴酒造, 菊正宗酒造, ...). Oh, I didn't even know that キリンビール is still officially 麒麟麦酒株式会社.
It is because the original inventor of Toyota was a man by the name of Kiichiro Toyoda with a D. Both the Katakana and Hiragana with the ten ten for the Da sound make for the name to have 10 strokes. In Japanese the number 10 or 十 in Kanji has the connotation of being at a cross roads. While the number 8 or 八 in Kanji is lucky because it widens at the bottom which reminds one of prosperity and growth. Because Toyota is not technically a correct word in Japanese it is therefore written in Katakana.
PS I work at Toyota