It is quite surprising what you see being played for Morris but there are a number of common instruments which are regularly used. This is not meant to be a definitive list but just a guide.

The Melodeon. This is one of the common Morris instruments - it has been around since the 19th century and was a development of the mouth organ (with bellows!). It is a diatonic instrument- it plays in a set key rather than chromatic like a piano accordion which can be played in any key. Melodeons come in many varieties, one two or three rows ( one key to each row ) some with 'accidentals' which in practical terms means the notes you can't get on the main row are in a separate row on their own. Morris melodeons usually come in D/G which means the outer row is in the key of D and the inner row is in the key of G. Each button plays a different note when the bellows are pulled than when they are pushed which makes for an interesting time but is great for music with 'lift'. Sometimes you can find a note on the other row but often you end up leaving it out! The left hand keyboard has a number of buttons covering bass and chords in the key(s) of the right hand keyboard - this limitation can give a good rhythm but is often overdone giving a 'pedestrian' sound! Melodeons are produced in a number of countries but you are most likely to see German or Italian instruments, though there are English, French and Irish versions in various keys available. The Anglo Concertina. Basically, the Anglo version is a diatonic instrument, a bit like a 'folded' melodeon ( not literally! ) where each button can play two notes, one on the push and one on the pull. The lowest notes start with the left side and the highest on the right. The English Concertina. This concertina is chromatic and each button plays the same note on both push and pull. Most concertinas date from the early part of the 20th century or earlier and can be valuable instruments. The two main makers where Wheatstone and Lachenal but there were others; the history is quite fascinating, and is worth investigating if you are interested. There are still makers today, Hohner and Gremlin names are often seen. English concertinas didn't just come in the size you see generally, there were different sizes and tuning, such as baritone, often used in the concertina bands that were popular during the mid part of the 20th century. Other concertinas/general.
There are other variations of concertina with different button layouts and tuning as well as wood or metal ends, bone or metal buttons, brass or steel reeds. Since many of these instruments were made at the end of the 19th century and are normally well looked after, they can be quite expensive to buy.

The Accordion. The piano accordian is a familiar instrument and is common in the folk tradition. It has the major benefit of being capable of covering any key. There are a number of sizes, the larger ones having more notes, particularly for the left hand giving a greater flexibility of sound. The worst part of the accordian is its weight - just ask a Morris man carrying one around. These instruments are mainly German or Italian and in various sizes, normally referred to in terms of the number of left-hand buttons , e.g. 80 bass.
The button accordion, common in France, is similar but instead of piano keys, the buttons are just that and as such they are often confused with the melodeon. However, the button accordion is a chromatic instrument just like the piano accordion but with the buttons laid out in rows the playing style is totally different and suits the French style of music. It can be used for the Morris but is much less common. The Pipe and Tabor (Whistle and Drum). This is one of the old Morris instruments and is highly prized, when played properly, for keeping time and the men under control. The pipe has three holes, one underneath, unlike the more usual 'tin' whistle and so is a little restricted in range - it is tuned to one key. This is why pipe players tend to carry a number of pipes in various keys. The Pipe and Tabor has its historical roots in the Military. The Violin/Fiddle. This is an excellent Morris instrument in the right hands, musical and capable of 'drive'- it is also chromatic so can play in any key. Its main disadvantage is that it is not very loud outside and for this reason you often see the violin attached to a small amplifier. The folk fiddle is one of the most popular instruments in folk bands - Thomas Hardy was a highly talented fiddler. Other instruments.
Instruments seen playing for the Morris include the most diverse range you can imagine but generally, these are used in addition to the other instrument(s) as backing or as a feature. These include electic guitars, trombones, saxaphones, triangles, tubas, comb-and-paper and even a hose-pipe with funnel!! Not for the faint-hearted.