South America is a paradise for cycling and also for mountaineering. It’s also a great place to combine the two activities.

Gear
Depending on your ambitions you have to carry a lot of technical gear or nearly nothing (apart from your cold weather cycling gear). There are many 6000+ peaks without snow cover, so you don’t need specific mountaineering gear. This can be handy if you like to travel light and mountaineering doesn’t have a very high importance on your journey. 

If you like to climb ice-covered peaks, you’ll likely end up with a lot of technical equipment. Although in many places in South America you can easy send your luggage by bus to the next big city and travel light in between.

We didn’t carry specific mountaineering gear (apart from hiking boots) and for us it was ok.

Where to buy mountaineering gear
In big cities like La Paz, Huaraz, Mendoza, … you’ll be able to buy some sort of mountaineering gear. Argentina and Chile also have good and reasonable priced online shops. But if mountaineering is an important part of your journey, you should bring your gear from back home.

Peaks
Just a few ideas, but definitely not a complete list

Ecuador
The two big volcanoes Chimborazo and Cotopaxi are both ice-covered and not super-easy to climb.

Peru
Cordillera Blanca: Huaraz is the mountaineering hub for the Cordillera Blanca. Most peaks over 5000m are ice-covered. There are plenty of local operators and Huaraz has a few good outdoor and mountaineering shops.
Around Cuzco: We rode around Ausungate, but didn’t climb it. There are definitely a few mountains to climb.
Around Arequipa: Chachani is just about 6000m and one of the easiest 6000m-peaks to climb worldwide. Usually no need to cross snow- or icefields. Don’t underestimate the altitude. Arequipa is just 2600m and most operators make the camp on 5100m.

Bolivia
There are a few peaks around La Paz. Most 6000+ peaks are ice covered.
South Lipez: Uturuncu is just 6008m high and very popular with cyclists, because it’s possible to ride a bike up to 5700m!
Sajama NP: There are a few quite easy to climb volcanoes around Sajama. It’s possible to organize guides, transport and equipment in Sajama village. The volcano Sajama itself is not that easy to climb. (requires ropes, etc)

Chile
Vicunas NP: There a few easy to climb volcanoes around Vicunas NP
San Pedro di Atacama: It gets drier down here… Also in this region lots of easy to climb volcanoes. But water is scarce and sometimes you can’t ride/drive higher than 4000m, what makes a long climb by foot.
Los Seis Miles: There are countless options in the region of Ojos de Salado. Make sure you bring enough warm clothing and a windproof tent.
Further south: no idea…

Argentinia
Around Salta: We tried to climb Llullaillaco. It’s a doable peak, but don’t underestimate its isolated position. It’s very fare away from everything else…
Los Seis Miles: You can approach the area also from Argentina and not only Chile. The Argentinian side is much wilder and not so easy to access.

A few bike & climb peaks we can recommend
Chachani 6075m, Peru/Arequipa
Uturuncu 6008, Bolivia/South Lipez
Pili 6048m, Chile/San Pedro di Atacama

Permits
In Chile you have to get a permit for many mountains. Usually it can be obtained for free, but it takes some time. Plan ahead…

In Argentina you have to register (free) if you plan to climb Ojos de Salado: http://www.turismo.catamarca.gob.ar/registro-obligatorio-de-expedicion/
If you plan to climb Aconcagua you have to pay a hefty fee (depending on the season)