My family is of Dutch origin and all my life I have studied the Dutch trying to learn about my heritage. I'm also a big fan of all things Japanese so I'll try to explain this the best I can:
Yes, "morumotto" (written モルモット) is a loan word from the Dutch "marmot". Marmot is now obsolete in the Dutch language when describing a guinea pig.
During the Japanese Edo period (1603 to 1867), the Dutch East India company began to trade with Japan, probably around 1609. The Dutch traded both in Hirado and a few years later in Dejima. Both of these are located in Nagasaki Prefecture.
(Side note: 1609 is the same year the Dutch East India Company commissioned Henry Hudson to sail around Manhattan, which the Dutch West India Company would soon officially settle as Nieuw Amsterdam in 1624 - now known as New York City!)
During the Edo period, Japan enacted a self imposed isolation, known as "sakoku". This limited the foreign trade that was allowed to take place between Japan and foreigners. There are several explanations for this with the most widely accepted being that Japan saw the empires of Spain and Portugal a threat to their political stability and religion. Basically, sakoku was a means of control.
So up until the 1850's, the Dutch were the only Westerners (meaning Europeans) allowed to access Japan - and even then so - they were limited to Hirado and Dejima. Other Europeans that landed on Japanese soil were put to death. Other countries, such as China, also traded with Japan during this period, but overall, the Dutch East India Company was the main trader from the West with Japan for almost 250 years!
Additionally, the Japanese learned about western science, art and literature from the Dutch and termed them "Dutch studies". It was only natural that Japanese words of Dutch origin started to develop.
With that explained, the Dutch term "marmot" is no longer used and has become obsolete. Today, the Dutch use "guinees biggetje" (literally Guinea piglet or Guinea porkling) or "cavia" or sometimes "huiscavia" - this adds "house" to "cavia". To describe a lab rabbit or lab guinea pig, you might use "proefkonijn".
There are other Japanese words for guinea pig besides "morumotto". They also use "ginia piggu" (written ギニアピッグ) and "tenjiku-nezumi" (written テンジクネズミ).