I would like to know whether a Japanese word combining at least two kanji can combine both on'yomi and kun'yomi characters, within the same word.
The answer is yes. There are "重箱読み (ju-bako-yomi)" and "湯桶読み (yu-tou-yomi)" in Japanese vocabulary (in 2 kanji words only).
重箱 is read as Ju-bako. "Ju" is on'yomi and "bako" is kun'yomi. Like this, it is called 重箱読み when 1st kanji is on'yomi and 2nd kanji is kun'yomi.
湯桶 is read as Yu-tou. "Yu" is kun'yomi and "tou" is on'yomi. Like this, it is called 湯桶読み when 1st kanji is kun'yomi and 2nd kanji is on'yomi.
王様(ou-sama) : on + kun --> 重箱読み
味方(mi-kata) : on + kun --> 重箱読み
丸太(maru-ta) : kun + on --> 湯桶読み
見本(mi-hon ) : kun + on --> 湯桶読み
Sii's answer is a good one.
To add a clarification, you asked:
Can a Japanese word combine both on'yomi and kun'yomi characters?
Japanese characters are neither on'yomi or kun'yomi. The characters are just graphical representations.
The words on'yomi and kun'yomi describe the readings for Japanese characters (specifically kanji). That's actually what the yomi part means: it literally means "reading".
For kanji, most characters have multiple readings: usually at least one on'yomi, and usually at least one kun'yomi.
- The word on'yomi is spelled 音【おん】読【よ】み, literally meaning 音【おん】 (on, "sound") + 読【よ】み (yomi, "reading"). The on'yomi pronunciation of any kanji usually derives from Middle Chinese, as the Japanese approximation of the sound of the Chinese word when it was borrowed from Middle Chinese over 1300 years ago.
- The word kun'yomi is spelled 訓【くん】読【よ】み, literally meaning 訓【くん】 (kun, "meaning") + 読【よ】み (yomi, "reading"). The kun'yomi pronunciation of any kanji usually derives from native Japanese vocabulary, as the Japanese approximation of the meaning of the Chinese word when it was borrowed from Middle Chinese over 1300 years ago.
1300+ years is a long time, and some things have changed over that period. Some kanji are even relatively recent inventions, such as 腺【せん】 (sen, "gland"). And language is always irregular. There are consequently exceptions here and there, such as kanji that have only on'yomi and no kun'yomi (like 腺【せん】), or kanji that have only kun'yomi and no on'yomi (like 畑【はたけ】 [hatake, "dry cultivated field"]). There have also been sound shifts in both Chinese and Japanese, so the Chinese word 十 is pronounced as shí in Mandarin, but as jū in Japanese -- these both derive from Middle Chinese pronunciation //d͡ʑiɪp̚// (roughly similar to the pronunciation of the English word Jeep).
To summarize, remember that Japanese characters have yomi, but the characters themselves are not yomi.